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international Workshop —
on Low-Frequency
Propagation and Noise

Volume 1

Woads Hole, Massachusetts
14-19 October, 1974

1977

Sponsored by CNO (OP-095)
Supported by CNR

Coordinated at the
KAaury Center for Ocean Science
Department of the Navy
Washington, D.C.

wt
0 0301 O0?bO0ee 4

International Workshop
on Low-Frequency
Propagation and Noise

Volume 1

Woods Hole, Massachusetts
14-19 October, 1974

Supported by CN

pocumer
eeordinated at ihe see”

Maury Center for Ocean Science
Department of the Navy
Washington, D.C.

Sponsored by CNO oF PAno

ACKNOWLEDGEMENTS

These proceedings record the first plenary session of the
International Workshop on Low Frequency Propagation and Noise,
sponsored by the Director, Antisubmarine Warfare Programs of the
U. S. Navy Chief of Naval Operations staff. Scientists from six
nations--Australia, Canada, New Zealand, Norway, the United Kingdon, .
and the United States--participated. The Workshop was conducted
by the Chief of Naval Research. The executing agency was the Maury
Center for Ocean Science; Director, Dr. J. B. Hersey. Commander A. G.
Brookes, Jr., USN, coordinated the conduct of the symposium which
was held at the Woods Hole Oceanographic Institution, Quisset Campus,
Woods Hole, Massachusetts. The Office of Naval Research is indebted
to the Institution for its excellent hosting of this meeting and in
particular to Mr. Charles S. Innis for his conspicuous efficiency
and skill in making all local arrangements. All participants in
the Workshop made substantial contributions either by preparing the
papers listed in the Table of Contents or by chairing various sessions.
These proceedings were recorded in detail by Ace Federal Reporters,
Inc. The completeness and quality with which the proceedings were
recorded are much appreciated and have made possible the level of
detail these proceedings contain. The proceedings were edited
primarily by the authors themselves and by Mr. F. P. Diemer in the
Office of Naval Research and Commander A. G. Brookes, Jr., USN. The
final editing and preparation of these proceedings were performed by
Science Applications, Inc., under contract to the Office of Naval
Research. Technical editing was performed by Drs. J. Czika, J. S.
Hanna, and R. C. Cavanagh under the direction of C. W. Spofford,
all of Science Applications, Inc.

The illustration on the title page displays iso-loss contours in range and depth generated
by the Parabolic Equation Model of Dr. F.D. Tappert. Regions of heaviest shading correspond
to losses of less than 80 dB re 1 yard, lighter shading to losses between 80 and 90 dB, and
lightest shading to losses greater than 90 dB. The calculation is for a constant (pressure)
gradient sound-speed profile in water 16,000-feet deep overlying a highly absorbing bottom.
The source is at a depth of 8,000 feet, the acoustic frequency is 50 Hertz, and the maximum
range is 50 nautical miles.

Details on the technique and more examples are contained in Dr. Tappert’s paper entitled
“Selected Applications of the Parabolic-Equation Method in Underwater Acoustics’? found in

Volume I of these Proceedings.

Title page was designed by Frank Varcolik, SAI.

PREFACE

The International Workshop on Low-Frequency Propagation and
Noise was held at the Woods Hole Oceanographic Institution, Woods
Hole, Massachusetts, from October 15 to 19, 1974. These Proceedings
consist of either author-supplied texts or edited versions of the
oral presentations and edited condensations of the discussions. In
the edited sections, the editors have made every effort to render
faithfully the essential content of the oral presentation or dis-

cussion.

These Proceedings are presented in two volumes, each consisting
of 2 days of presented papers. Several of the original presentations
have been superceded by a published version, which the authors also have
submitted for publication here. In these cases, with the permission
of the authors and the publishers, the published articles are
reproduced here in facsimile. The presentations so rendered are

the following:

i) Dr. Weinberg's paper appeared as NUSC Technical
Report 4867.

ii) The presentation of Drs. Flatté and Munk contained
some of the information presented in the three
articles published in the Journal of the Acoustical
Society of America.

iii) Dr. Raisbeck's paper appeared in the U. S. Navy
Journal of Underwater Acoustics.

iii

INTERNATIONAL WORKSHOP
ON LOW-FREQUENCY PROPAGATION AND NOISE

VOLUME I

TABLE OF CONTENTS

Page
I. ENVIRONMENTAL MEASUREMENTS (J. B. Hersey, Session Chairman)
TNTRODUGLION, DG dice Bien gHELSCUl. ts ys) «ie ico esl el ne ear 6 Hl
TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE

OHA A IGS mito tai ebublalisoyee an ee 6 oo oro oo mac.6 6 5 oo fo Wf
THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES,

Exemner Arete Gli Sit1'cl ian vel Mota cl et its? ell le) sed ey pisiiiasr ue. ie: utst eis) ter nell oe 33
EXPLOSIVE SOUND-SOURCE STANDARDS, M. S. Weinstein ..... 61
APPLICATION OF RAY THEORY TO LOW-FREQUENCY PROPAGATION,

yan) WMasliglyesaey “GG G85 6 0 8 GO G6 oo o o oo Oo @ oo oe 93

II. PROPAGATION THEORY (R. R. Goodman, Session Chairman)
NORMAL MODES IN OCEAN ACOUSTICS, D. C. Stickler ...... 125
SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD

IN UNDERWATER ACOUSTICS, Frederick Tappert ...... . 155
CALCULATION OF THE EFFECT OF INTERNAL WAVES ON OCEANIC

SOUND TRANSMISSION, Stanley M. Flatté and

BGC Cr Chase ela PDCTG. ser i 6 sale 1s ee wee 2 ce ee ZO
SOUND PROPAGATION THROUGH A FLUCTUATING STRATIFIED OCEAN:

THEORY AND OBSERVATION, W. H. Munk and F. Zachariasen. . 211
INTERPRETATION OF MULTIPATH SCINTILLATIONS ELEUTHERA TO

BERMUDA IN TERMS OF INTERNAL WAVES AND TIDES,

Freeman Dyson, W. H. Munk and B. Zetler. ........ 233

Teele

IV.

VOLUME I

TABLE OF CONTENTS (Cont'd)

EFFECTS OF BOUNDARIES (M. Schulkin, Session Chairman)

ACOUSTIC PROPERTIES OF THE SEA FLOOR, John Ewing. ....

THE EFFECT OF ROUGH INTERFACES ON SIGNALS THAT PENETRATE
THE. BOTTOM Ger Wien HOREOM; 10 Ti. Cais) Bese a GsyT ON Save nol neaerenecs

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION,
F\DUNGCP Ie glievolepekfor 5 “SG 6 6% 6 oO 6 G& @ oa oto 5S 6 EG

FORWARD SCATTERED LOW-FREQUENCY SOUND, W. I. Roderick ..

SIGNAL CHARACTERISTICS (T. G. Birdsall, Session Chairman)

COHERENGE), = LACOdOre) Gen BLLAS ale Men we) tou rene testes Neel ves nat <ok et oe
HLUCTUATIONS:©— VAN -OVERVEEW) cirar DUCT. teml sy cc tele ©
SOUND PROPAGATION IN A RANDOM MEDIUM, Robert H. Mellen. .

PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES,
oa Gyesbeella Go G6 6 Oo 6 6 6 G 6 oS S&S G Se 8 6 o 6 6

FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND
DOPPLER, SPREADING, (Hi Al Debherrart ens s.2) 2 <) 2 ense

DESIGN OF TRANSMISSION LOSS EXPERIMENTS, J. S. Hanna. . .

Page

249

20D

299)

325

385i)

365

387

423

465

509

INTERNATIONAL WORKSHOP
ON LOW-FREQUENCY PROPAGATION AND NOISE

VOLUME II

TABLE OF CONTENTS

Page
V. SPECIAL EFFECTS ( R. H. Nichols, Session Chairman)
SURFACE DECOUPLING EFFECTS, M. A. Pedersen, D. F. Gordon,
ANG De, WHBEC! miu ea ce tee ence iewiis: We ne ie. Gell%s oboe yy) ep et eben 559
A THEORETICAL APPROACH TO THE PREDICTION OF SIGNAL
FLUCTUATIONS DUE TO ROUGH-SURFACE SCATTERING,
Ree Memeiablancarand ts Yao Hala DCL cama is, eit sie ey te ei ie) tens 583
SOFAR PROPAGATION OF WIDE-BAND SIGNALS TO LONG RANGES,
RE ae OTa CT MMC, (a) maces ter tier efeaiey wastes io Melina aoe He: Ene ce) Rome eine ale Bey ie 633
CONVERGENCE ZONE DEPENDENCE ON FREQUENCY, R. M. Fitzgerald. 667
VI. GEOGRAPHIC EFFECTS (E. E. Hays, Session Chairman)
LOW-FREQUENCY PROPAGATION IN THE ICE-COVERED ARCTIC OCEAN,
Heng Wie ehUESCHALC: va) es) et) et ler er fe, ep ue) te) le Se se ies 683
ARCTIC ENVIRONMENTAL LF ACOUSTICS MEASUREMENTS, MODELS
ANDMEEANS:) yBeaumome May (BUCK. seis seu er ter re) eel ie) cee) Ge reo 725

ENVIRONMENTAL FACTORS AFFECTING LOW-FREQUENCY PROPAGATION
UN} BMGIa) OlelayAN( 5 Ielehvalfel (EG sehe\ipeblielei, FG & G 6 oy oo Goo oO 769

vii

VOLUME II

TABLE OF CONTENTS (Cont'd)

Page
VII. NOISE MECHANISMS (I. Dyer, Session Chairman)
AMBIENT—-NOISE MODELS, ReaG.Gavanaghin. i. ws ivet (5. te meeunee 801
GEOGRAPHICAL VARIATION OF AMBIENT NOISE IN THE OCEAN
FOR THE FREQUENCY RANGE FROM 1 HERTZ TO 5 KILOHERTZ,
Robert L.. Martin and Anthony di. Perrone . . 6 ss = «=. 817
STATISTICAL ANALYSIS OF SHIP-GENERATED NOISE,
Sec Cig DAUD Dia aise See PRO ar coPes wloPM cel is GS aces: Pay ole 843
VIII. NOISE MEASUREMENTS (W. A. Von Winkle, Session Chairman)
VERTEGAL NOLESE) DESTREBULEON) sic ComANAGLSON ii. eens caine 859

DIRECTIONAL NOISE AMBIGUITY RESOLUTION WITH LINEAR
ARRAY Sis *GOLAOMMRaATS DECK meme ueetl cy en ok on co atu tody cs ol Rem ours 887

vili

LOW-FREQUENCY PROPAGATION AND NOISE WORKSHOP

INTRODUCTION

Dr. J. B. Hersey

Office of Naval Research

It is a great pleasure to welcome members of the Low-Frequency
Propagation and Noise Workshop to this, its first meeting. This inter-
national workshop is sponsored by the Director of Antisubmarine Warfare
of the staff of the U. S. Chief of Naval Operations. It recognizes
the growing cooperation among the participating nations in application
of low-frequency underwater acoustic systems in ASW. Also, it is
closely related to a series of workshops sponsored by various elements
of the U. S. Navy oceanography and undersea warfare community. The
broad purpose of the U. S. workshops is to support progress toward
solving the priority problems of the U. S. Navy in undersea warfare

and other concerns of the Navy where the oceans are influential.

The objective of this workshop is to assess our understanding of
low-frequency ocean acoustics and to identify and prioritize what
programs of investigation should be emphasized in the next 5 to 10
years. It is intended that our final product will be proceedings of
this meeting and a planning guide that can be a useful and influential

instrument for all nations here convened.

The general approach is first to hear reviews of as comprehensive
a series of topics as possible during the next 4 days. Most of the
talks are intended strictly to provide a basis for assessing our under-
standing and knowledge of this field. The remaining few talks present
viewpoints that are thought to be of special interest because they

represent new departures. We have tried to order the program so that

HERSEY: INTRODUCTION TO LOW-FREQUENCY PROPAGATION AND NOISE WORKSHOP

there will be ample opportunity for discussion, all of which will be
recorded. On Saturday morning, we will make decisions on a structure
of small working groups that will be responsible in the next 6 months
for preparing a written assessment and recommendations for future
programs of investigation. My office will make every effort to speed
the availability of the proceedings of this meeting to its members so
that all the material presented and discussed here will be available
to the working groups. These proceedings will also be published and

suitably distributed.

I earnestly hope that all members of the workshop will participate
to some degree in the main work of the workshop — that of the next
6 months. You have received as part of your registration package two
questionnaires. One is to be filled out and handed to CDR Brookes
this morning. It will serve as the basis for the first cut at
organizing the working groups. The second, intended as a guide to
the Steering Committee, is to be filled out no earlier than Thursday
afternoon so as to be available for the meeting of the Steering
Committee Thursday evening. You will notice that it gives you an
opportunity to change your mind about the first questionnaire. It
also gives you an opportunity to comment on any aspect of the objec-
tive of the workshop, its content so far, and what you think should
be done by the time the workshop is disbanded. Please use it

generously.

The final plenary session of the workshop is now planned for
May 1975, either in San Francisco, Monterey, or San Diego. Its
objective will be to hear, discuss, and make provision for rewriting,
editing, and publishing the planning guide. It is envisioned that
talks will be presented describing the content of the several chapters
as determined by the working groups. Further, a major part of this

session will consist of discussion of this material, which should be

HERSEY: INTRODUCTION TO LOW-FREQUENCY PROPAGATION AND NOISE WORKSHOP

available to all members of the workshop well before the meeting. A
small editorial staff will be responsible for final editing, produc-
tion, and distribution of the planning guide. A record will be kept
of the discussions at the second plenary session, but no useful

decision can be made at present about its publication.

So much for generalities. In the next 4 days we will be review-
ing some things old and some things new that represent our partial
understanding of the characteristic behavior of low-frequency sound
waves in and below the oceans. Low frequency here means the range
from 1 to 1,000 Hertz. A quite arbitrary range which, unfortunately
perhaps, includes at low frequencies the Airy wave of the deep ocean
basins and at high frequencies phenomena that are sensitive to rather
fine details of water structure and ocean floor topography. The past
emphasis in research and applications is most uneven. Major U.S.
emphasis has been on the spectral region from 20 to 150 Hz with some
far less intense emphasis on the region from 150 to 1,000 Hz. Only in
the past 3 or 4 years have we attempted anything significant below 20
Hz. Thus, we will find - if we look - that there is great unevenness
in our information throughout this spectrum. In some areas of interpre-
tation, there is great scope for speculation, because there is so
little hard data; whereas in others we have so much information that
the knowledgeable interpreter may feel tongue-tied. In the next
6 months — and a lot longer — we should look at the spectrum
encompassing both extremes in order to learn what is going on in the
ocean, thus solving many of the practical problems of warfare there —

and other marine concerns of mankind as well.

All nations represented here use their knowledge of ocean
acoustics more or less intensely for some or all of the following

purposes:

HERSEY: INTRODUCTION TO LOW-FREQUENCY PROPAGATION AND NOISE WORKSHOP

1) To forecast the performance of existing sonars over
the next few hours, tomorrow, next week, next year,
and so on

2) To analyze operations or operational exercises as a
means of improving system performance

3) To analyze the results of acoustic intelligence
4) To assist in force level trade-off studies
5) To assist in the identification and selection of

new systems design options

6) To assist in the entire development process after
options have been selected for development.

In formulating objectives within the framework of these purposes, we
are driven by scientific or technical opportunities and constraints

in the face of the potential enemy's capabilities and characteristics.

Our investigations can be programmed either to provide a tech-
nology base on which new analysis tools or new sonar systems can be
developed or they can help develop a needed capability. The U.S.
Defense Department has long subdivided these efforts by names such
as research, exploratory development, advanced and engineering

development, and so on.

The non-U.S. participants will inevitably hear American partic-—
ipants refer to these activities by their number, the budget sub-
elements 6.1, 6.2, 6.3, and so on. In principle, I believe this work-
shop to be concerned with 6.1, 6.2, 6.3, 6.4, and 6.6. In practice
we hear little of 6.6, and so far as I am aware, have no programs
whatever in 6.4. For the remainder, I find it easier to divide our
concerns into 6.1, 6.2, and 6.3 as follows. In 6.1, we should study

acoustic and oceanic processes and how they interact. In 6.2,

HERSEY: INTRODUCTION TO LOW-FREQUENCY PROPAGATION AND NOISE WORKSHOP

we should examine how the resulting understanding of ocean acoustics
can be applied in the framework of the broad purposes stated above.

In 6.3, we determine as precisely as deemed necessary the significance
of ocean acoustics in supplying a particular fleet service, analysis
tool, or support for the development of a specific new sonar system.

Again, all of these are a proper concern in this workshop.

In making our assessment of present understanding of low-frequency
acoustics, significant confusion may result from the unevenness Of our
understanding. Acoustical theory has long been able to modél complex
processes for some simple configurations of the sound medium, and
the question has been raised repeatedly whether much, if anything,
remains to be done in basic acoustics. Nevertheless, the ocean, its
surface, and its floor are so complex that these models are of limited
practical use. The last 5 to 8 years have seen an intense effort in
the U. S., mainly in 6.2 and 6.3 programs to develop models that would
deal in practical and useful detail with the major complexities of
the ocean and predict transmitted sound levels and noise. We have
depended altogether on modern digital computing techniques and on
comparisons with measurements. These modern methods are only now
beginning to teach us when and where sweeping simplifications of the
shape of the boundaries and the acoustical properties of the ocean
are both useful and adequate. We shall be looking at some of these
results. How should these computational methods be developed in the

future?

We have done surprisingly few strict comparisons of acoustic
measurements with model analysis based on simultaneously measured
acoustic data and oceanic properties. The necessary impact of the
few comparisons available has not had time to be fully felt and
digested. Even so, important lessons are emerging. Nevertheless,
we still don't now know how detailed a program of measurements is
required. I hope that the workshop can help us chart a good course

to answer the nagging question of: How much is enough?

TIME VARIATIONS OF SOUND SPEED OVER
LONG PATHS IN THE OCEAN

G. R. Hamilton
Ocean Science and Technology Division

Technology Division

From 1961 to 1964 a series of precisely located and timed

SOFAR charges were fired off Antigua to measure the trans-

mission time stability of the sound channel axis arrival

(i.e., the SOFAR signal cutoff) to MILS hydrophones at

Eleuthera, Bermuda, the Canary Islands, Barbados,

Ascension and Fernando de Noronha. The sound transmission

speed was found to be stable for a few hours but it

could not be predicted a week in advance. An application

to the precise location of missile impacts using SOFAR

signals, based on the dropping of SOFAR charges at the

missile impact position within a few hours of missile

launch, is described.

The most extensive measurement of sound-speed variations over
long distances were made in a program in the early 60's called SCAVE,
for Sound-Channel Axis Velocity Experiment. The locations of the ex-
periment are shown in Figure 1. Results were published in the proceed-
ings of the Naval Underwater Acoustics Symposia in 1962, 1963, and

1964.

The measurements were designed to make it possible to use SOFAR
charges to determine the accurate impact position of Polaris missiles
launched southeast into the Atlantic from Cape Kennedy, Florida. With
a range of about 1600 miles, these missiles impacted in the open ocean
east of the Caribbean. For this flight range, they could not be tar-
geted to impact close to an island or coast line where shore-mounted
radars or shore-connected bottom hydrophones could be used for deter-
mining impact position without overflying islands. Could SOFAR charges
carried in the missile be used to accurately locate the impact position
at a mid-ocean location? What impact position accuracy would SOFAR

charge provide?

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

ATLANTIC MISSILE RANGE
MISSILE IMPACT LOCATION SYSTEM

<
: 40°
AZORES :, ye
BERMUDA
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1A.
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Figure 1. LOCATIONS OF SCAVE

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Our initial ideas were to drop SOFAR charges over accurately lo-
cated bottom hydrophones off Antigua, time the SOFAR arrivals 800 miles
away at Bermuda, and basically use this experimental sound-speed mea-
surement to calibrate sound velocimeters, in absolute terms, for the
ocean conditions at 3000- to 4000-ft depth and 3°c. This was the only
method we could conceive to obtain an absolute sound velocimeter cali-
bration for these temperature and pressure conditions. For example,
in 1962, when calibrating a sound velocimeter in the laboratory, there
was an elusive one-foot-per-second difference in various tabulated
values of sound speed for distilled water at surface temperatures

and pressures.

In planning the experiment, we assumed that the axis sound speed
at any open-ocean location would be stable. We would use this 800-mile
travel-time measurement to calibrate the velocimeters in absolute terms
based on multiple lowerings along the transmission path. Since SOFAR
charges off Antigua could also be received on the MILS (Missile Impact
Location System) sound-channel axis hydrophones at Eleuthera, at
Fernando de Noronha off Brazil, at Barbados, and at Ascension, we
recorded on these as well. Looking ahead, since our ultimate problem
was to accurately locate a missile SOFAR charge in mid-Atlantic, a hy-
drophone was obviously needed in the northeast Atlantic to balance
any unknown bias from the existing MILS hydrophones to the south and

west. Such a hydrophone station was installed in the Canary Islands.

Figure 2 shows a typical SOFAR signal received over a Sargasso
Sea transmission path. Typical, in that for a 900-mile transmission
path the signal has a 9-second duration and terminates with a sharp

cutoff. For this hydrophone buoyed up into the sound-channel axis,

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

VELOCITY SOFAR TRANSMISSION PATHS
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Figure 2. TYPICAL SOFAR SIGNAL OVER SARGASSO SEA
TRANSMISSION PATH

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

the signal-amplitude cutoff is in excess of 30dB and occurs in about

0.02 seconds. The trace shown is a Sanborn direct-writing hot-pen re-
corder in the log-audio mode (i.e., the log of the rectified audio sig-
nal). The ray diagram shown is from the original Ewing and Worzel 1947
SOFAR paper. For this Sargasso Sea sound-speed profile with the Sargasso's
500m thick, near surface layer of 18°C water, the first arrivals of the
SOFAR signal travel along paths that are near bottom grazing and

through the 18°C near-surface water.

In Figure 3 is shown the bottom hydrophone array off Antigua over
which SOFAR charges were dropped. The water is 3,000 fathoms deep.
Three hydrophone signals are needed to locate and time an underwater ex-
plosion. With six hydrophones in this array, there is redundant data
for greater system reliability and for greater time and position accuracy.
One SOFAR charge could be located relative to another in the central
area of this array with a precision of 30 feet. This shot-position pre-
cision on a transmission path of 1,000 miles to a fixed hydrophone
means the error in the relative sound-speed measurement due to source-

charge positioning errors is of the order of 0.04 feet per second.

In our SCAVE tests, and we ran about 25 or 28 of them, we
chartered a small boat in Antigua as a SOFAR charge drop boat. This
boat was the type of yacht you could charter for about $2,000 a
week. Normally a one-week charter was required to set up aboard,
sail to the hydrophone area and drop SOFAR charges for two hours,
return and offload. On one SCAVE, we dropped SOFAR charges every
hour for 24 hours, and on another occasion every hour for eight

days.

Figure 4 is a typical record for an overhead SOFAR signal on
these bottom hydrophones, illustrating the S/N ratio and system fre-

quency response that made the 30-ft shot-position precision possible.

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

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HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Again, this is a Sanborn recorder with log-audio recording. The verti-
cal scale for each of the records covers 50 dB. Shown at the bottom

is a one-second interval. Direct signals are marked with a D,

surface reflections with an S. Rise times for these signals are

less than 0.01 seconds. The two pulses following D and S are sub-

bottom echoes.

In Figure 5 are shown the results of the first year's program.
Primarily shown here are results for two phones at Bermuda and three
at Eleuthera. The error bars indicate the full spread of the sound-
speed data. There are two immediate conclusions. Obviously, the sound-
channel axial speed was not constant. There are times when the speed
remains constant for a month or two, but it can also change by 2 feet
per second within a month. The second conclusion concerned the cause
of the sound-speed variations. The month-to-month variations on the
Bermuda and Eleuthera phones do not correlate. The inference there-
fore is that the cause of the speed variations is not a phenomenon at
the source. Note also that speed variations at the two Bermuda phones
track very nicely. Although these phones are about ten miles apart,
the line between the "SOFAR Station bottom hydrophone" and the "BOA
Spd" (Broad Ocean Area suspended) hydrophone continues directly to the
Antigua hydrophone area. And so we inferred that whatever is causing
the variations in the average sound-channel speed between Antigua and
Bermuda is not some small-scale effect in the area of the receiving

hydrophones.

For the three Eleuthera phones, the transmission paths to each of
the individual phones are not identical, and the speed variations,

although similar, don't track as accurately as those at Bermuda.

In Figure 6 are the SCAVE results for 2.5 years. At the top are

three additional transmission paths, Ascension and Fernando de Noranha

TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

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HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

in the South Atlantic, and Barbados just 300 miles to the south of
Antigua. The obvious conclusions we reached from this were that we
couldn't predict the axis sound speed with the required accuracy for

missile-splash location.

Note the long-term correlation between Eleuthera and Bermuda.
For the first half of the program, there is a six-month cycle, rather

well displayed, that occurs four months later at Eleuthera than Bermuda.

Shown in Figure 7 are the results of an eight-day SCAVE with sig-
nals at hourly intervals. The dots are the actual sound speeds measured.
The dark line is a seven-point moving average. Although the sound-
speeds change over this eight-day period, they change slowly. Hydrophones
with closely adjacent transmission paths have similar changes. It was
from these data that we developed a system for using SOFAR signals to pro-

vide accurate missile-impact location estimates.

Essentially we calibrated or measured the axis sound speed for each
missile test for each receiving hydrophone. This was done by firing
SOFAR charges in the missile-impact area over bottom transponders which
had already been located. Ten SOFAR charges were fired before the
test and ten after the test. An average measured SOFAR speed for each

hydrophone was used for calculating the missile splash position.

In Figure 7, results from the Bermuda suspended phone for the
first 3 days suggest a sound-speed variation with the period of a semi-
diurnal tide. This Bermuda phone was at the 4000' sound-channel axis
depth, but buoyed 5000' off the bottom. It was not unreasonable to
suspect that this hydrophone moves back and forth with tidal currents
at Bermuda. Without data on this hydrophone's movements, it is there-
fore impossible to state whether this is an actual sound-speed varia-

tion or an artifact of hydrophone motion. The hour-to-hour variations

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here are of the order of 0.15 feet per second. The eight-day variation

is about 0.4 feet per second.

Figure 8 data are from a series of SOFAR shots across the center
of the Antigua hydrophone array on a NE-SW line to see if the axis sound
speed to each hydrophone was sensitive to small changes in the source
position. As shown, there is a small effect. Additional profiles would
have been required to determine if this was a source-position or trans-

mission-path effect.

Figure 9 is a record of the seawater temperature at the sound-
channel axis off Eleuthera. The equivalent sound-speed variations are
shown at the lower right. Shown are temperature variations in excess
of 0.5°C corresponding to maximum sound-speed variations of five feet
per second although more typical sound-speed variations are on the order

of three feet per second.

Figure 9 also illustrates a rough comparison of the Figure 6 data
for 1961, 1962, and 1963 Eleuthera hydrophone sound speeds with the
seawater thermistor temperature readings. Speeds from Figure 6 are
plotted as circles on Figure 9 with a vertical line drawn to the sea-
water temperature at that time. It is apparent that the correlation
between the two is rather poor, indicating that the sound speed at the
hydrophone is not the dominant factor controlling the average sound

speed over this 1,000-mile transmission path.

Figure 10 is a series of sound-speed profiles in the area of
the Antigua shot positions illustrating the extreme variability of the
water masses in this area at axis depths. Two velocimeters calibrated
to identical readings were used in this instrument package to increase

the confidence that the small perturbations in the profile were real.

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HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Figure 11 is a series of SOFAR signals before and after a missile
test as recorded at Bermuda. Shown are the scheduled detonation time,
the detonation depth, the difference in the propagation speed from the
average for the SOFAR timing tick marked, and the spread in range re-
sulting from this timing tick. These SOFAR signals do not have the
sharp, clean SOFAR cutoff of those fired over a flat bottom. Presumably,
forward scatter from the rough mid-Atlantic Ridge topography degrades

the cutoff for these signals.

Note that there is no variation in sound speed for the range of
detonation depths shown. Actually, this is what you would expect for
a SOFAR charge position east of Antigua. The average speed there at
the sound-channel axis is of the order of 4875 feet per second. At
Bermuda it's about 4890. So the axis is pinching down. This insensi-
tivity of average axis sound speed to detonation depth was very conve-
nient because the manufacturer making the SOFAR charges could never
meet a depth spec on the SOFAR charge. We thought depth variations
might cause speed variations and therefore wrote a 3 percent depth
specification. The manufacturer could never meet it, but, as it

turned out, it did not matter.

Summarizing the conclusions, 1) the sound channel axis speed was
not stable; we couldn't predict it, and 2) neither the sound speed at
the source nor the sound speed at the receiver seemed to control the
average speed variation over these long transmission paths. The speed
variations were apparently caused by what was happening in the water

masses between the source and the receiver.

TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

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HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

DISCUSSION

Dr. A. O. Sykes (Office of Naval Research): What do you read for

the cutoff on the records in Figure 11?

Mr. Hamilton: You go through a learning cycle to determine what
fits the data and gives nice results [laughter]. If the delta range
is large in data such as this, the sources of error are few: 1) the
shot boat position and the SOFAR detonation depth/time in the trans-
ponder array; 2) the SOFAR charge did not sink vertically; 3) the
SOFAR signal cutoff timing; 4) something in the physics of sound trans-
mission is not understood. It was never necessary to use either 2

or 4 to explain away inconsistent data.

Dr. D. C. Stickler (Applied Research Laboratory): Can you
speculate about the origin of the six-month period in your sound-speed

program?

Mr. Hamilton: There has been a recent series of papers by Jacobsen
of RPI in JASA discussing ocean Rossby waves and their effect on acous-
tics. Rossby waves are similar to the 200nm diameter eddies seen by
the Mid-Ocean Dynamics Experiment in 1974 between Bermuda and the
Bahamas. That six-month variation is about the period of these

eddies as they move westward a few kilometers per day.

Dr. Sykes: How far apart in time were the pre- and post-SOFAR
shots used to calibrate the missile impact area? And how closely do

the speeds correlate? Is it a matter of a day or so?

Mr. Hamilton: No, they were approximately two hours before and

after missile impact.

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Dr. Sykes: The reason for the question is I was trying to get an
estimate from you of how close acoustic and sound-speed measurements
should be made. That is, do you think a day apart is good enough?

Or a week apart? Or really a few hours? Must they really be simul-

taneous ?

Mr. Hamilton: You can see the individual variation from shot to
shot in Figure 1l. For this purpose the sound-speed measurement

didn't seem to be very critical.

But remember we aren't talking about the rays that are going
through the surface waters. We are talking about what is going
along the axis. The seawater temperature at Eleuthera on Figure 9
shows no correlation with the average axis sound speed to a nearby
hydrophone. The sound-speed measurement at a single site doesn't

seem to be important.

The fact that variations in sound speed to the two Bermuda phones
correlate so beautifully for transmission over exactly the same ocean
transmission path in Figure 5 and that these do not correlate exactly
to hydrophones on nearby paths for Eleuthera in Figures 5 and 6 leads
me to believe that what is happening at the source is not very import-

anicr

So I am coming to the conclusion that at least for this measure-
ment, the sound speed at either the source or the receiver is not
important for the average horizontal propagation speed along the

sound-channel axis.

Ms. E. A. Christian (Naval Ordnance Laboratory): It's no more

important than the rest of the path?

Mr. Hamilton: Right.

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Mr. M. A. Pedersen (Naval Undersea Center): I have several
comments on this particular presentation. I have worked up a number
of Pacific profiles for which the slowest arrival is not the axial
arrival. It seems that the slowest arrival is associated with the
steepness of the thermocline. That is, in one case I worked out the
slowest ray formed was about a yard-per-second slower than the axial
speed. And it corresponded to a ray which is turning around in the

steep thermocline portion.

Mr. Hamilton: Figure D-1l shows a comparison of some typical pro-
files and their SOFAR signals. For the Sargasso Sea profile from
the western North Atlantic, the slowest SOFAR arrival travels along
the sound-channel axis. The resulting signal is in the lower left.
In the typical Eastern Atlantic profile the Mediterranean outflow
broadens the sound-speed minimum and increases its value, so the
slowest arrival travels in the surface and bottom grazing ray paths.
This SOFAR signal is on the lower right. In the Pacific profile,
the SOFAR signal is shorter as shown in the middle bottom trace.

It is actually quite similar to what a western North Atlantic SOFAR
Signal looks like if you eliminate the rays that penetrate the 18°C
Sargasso water. In the Atlantic where you have the Sargasso water
in the surface 500 meters, this high sound speed near surface

water gives the early SOFAR arrivals in the lower left SOFAR signal.

I ran into the Eastern Atlantic SOFAR signals a few years ago.
When I saw these signals, I assumed they were from surface shots and
made some stupid statements to that effect. I then realized that
the Eastern Atlantic was a different ocean entirely with this entirely
different profile, that it just reverses the SOFAR signal completely

from the classical one Dr. Worzel published, 25 years ago.

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

SOUND SPEED

1463 m/sec 1493 m/sec 1524 m/sec
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SCHEMATIC SOFAR SIGNALS — 900 nm range
| | | Axis arrival
Rays that penetrate

into 18°C Sargasso

water
————_

WESTERN N. ATLANTIC EASTERN N. PACIFIC EASTERN N. ATLANTIC

Figure D-1. SCHEMATIC SOUND SPEED PROFILES

HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

Dr. R. P. Porter (Woods Hole Oceanographic Institution): Were
you able to resolve any other arrivals besides the axial arrival?
In other words, could you say there were specifically other paths
that stood out that might have been rays that were going through

water with a higher sound speed?

Mr. Hamilton: We occasionally would see it on the first group
of arrivals when we had a suspended hydrophone. Within a group we
would see the individual arrivals corresponding to the up- and
down-going paths on both ends. These were apparent for the earliest

orders, and then they start to get closer together, effectively over-

lapping.

Dr. Porter: Would you see this arrival structure then build up
in the final arrival? In other words, could you analyze those shots
in terms of the individual arrivals themselves prior to the axial
arrival, because if you could do that, then you could possibly look
at some of the average sound speeds through the remaining part of

the water column.

Mr. Hamilton: Most of our work was at ranges like a thousand
miles, and the cycling of the groups is about 30 miles, so we are
talking about 30 of the groups building up into the final peak.

We might see the first and the second, but after that it pretty much
ran together. It didn't look to us at the time like an interesting

problem to work on, and we didn't look at it.

Dr. W. B. Moseley (Naval Research Laboratory): In your data, the
temperature variability at neither the source nor the receiver
appears to directly correlate with the travel-time variability.
However, if you were dealing with the same water mass type throughout
the range, would you expect the statistics at either of the end

points to correlate with the statistics of the arrival time?

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HAMILTON: TIME VARIATIONS OF SOUND SPEED OVER LONG PATHS IN THE OCEAN

In other words, could you simply measure statistics at the end points,

or do you really need measurements throughout the entire path?

Mr. Hamilton: Are you asking whether one could take statistics
at the source and at the receiver and compute the average transmission

velocity?

Dr. Moseley: No, compute the statistics of the variability of

the average travel velocity.

Mr. Hamilton: Yes. However, in this problem statistics don't
help very much since the problem was: What is the actual range from
the launch pad to the impact point expressed as a deterministic,
accurate value? If the speed variation over that thousand-mile path
were plus or minus a foot and a half per second, it would equate to
a location uncertainty path of 1,800 feet. So we really weren't
interested in the statistics. We had to have as accurate a value

as we could at the time.

The Acoustic Output of Explosive Charges

Ermine A. Christian

White Oak Laboratory
Naval Surface Weapons Center
Silver Spring, Maryland

Although small explosive charges are widely used as
sources for underwater acoustics studies, a number of un-
resolved questions exist concerning the proper source
levels for use in data analysis. At the present time
there is no generally accepted "standard" set of source
levels, and deviations of 5 to 10 dB can be found among
published values. Better information is needed to define
a reference range beyond which finite amplitude effects
are negligible. Spectral energy levels are sensitive to
charge detonation depth, an experimental variable that
typically is not controlled in acoustic experiments.
These and other problem areas associated with explosion
sources are reviewed with comments on their quantitative
effects at low frequencies.

INTRODUCTION

I am well aware that the members of this acoustic community are
not, in general, entranced with the beauties of explosions physics.
It is a delightfully complex subject that is simply a pain in the neck
to someone who wants a neat, Simple source for underwater acoustics

research work.

Unfortunately, in today's sophisticated world we are trying to
do a number of rather closely controlled experiments. We are looking
for subtle effects, small differences, and we are looking for detailed
frequency dependencies rather than the broad-brush quantities that
sufficed a decade ago. This means we can no longer gloss over inherent
characteristics of the pressure waves generated by explosions, even

though they may be uncomfortably complicated. So let me be a purist

Sil

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

for a few minutes and talk about an explosion wave as it really is,

rather than as acousticians would like it to be.

Desirable attributes of sources for use in low-frequency acoustic
work are much the same, whether the sources are CW transducers or are
explosions. For use in the field it is desirable to have sources that
are reliable, inexpensive, with a high energy output at the desired
frequency range, and convenient and simple to operate under field test
conditions. For use in data analysis, it is desirable to have known
standard source levels, source level values that are compatible with
the sonar equation, and values that are predictable within some speci-
fied decibel allowance in a narrow-band frequency. Explosion sources
often come out ahead when considering field-use desirability, which
is why they are used so widely for underwater acoustics research. But
for the analysis end of the problem, explosions sometimes seem to be

intractable. Today I will show you some of the reasons this is so.

For today's discussion, let me use values of bandwidths and fre-
quency ranges and prediction errors that I have heard discussed within
the past year as being desirable in acoustics research work. (It may
be that during the course of this Workshop these values will be modi-
fied. If that is the case, I will only say that I hope all of them
will increase, from the point of view of our ability to utilize avail-
able information today.) We would like source levels in 1/3-octave
bands. And we want these levels to be predictable to within 1 dB,
over the frequency range of 10 to 300 Hz. Here I purposely use the
word "predictable," rather than "reproducible" to within 1 dB. Repro-
ducibility is not the problem with the explosive compositions usually
used in acoustics work. If you replicate the experiment — the charge

type and depth, the measurement point, the recording and analysis

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

systems — you reproduce the pressure-wave output within a few per-

cent.* Predictability, however, is another matter.

Given these quantitative constraints — source levels predictable
within 1 dB in 1/3-octave bands over the frequency range 10 to 300 Hz —
we are in rather poor shape for analyzing data taken with explosion
sources. We do not have known, standard source level values. We have
problems with sonar equation compatibility. And we very definitely

have problems with our 1 dB error allowance.

I don't think we need much discussion on the question of known,
standard values. Everyone in this room who has used explosion sources
is aware that among the published values in the literature, the in-
house publications, and the backs-of-envelope working papers we all
turn to, a wide assortment of source level values can be found. These
values vary perhaps by 5 dB, perhaps more, depending upon the band-
width of interest. In fact, the question is so wide open that I have
wondered if data reduction sometimes follows the line of "when in
doubt, blame the source level; and then look around until you find

one you like better."

To mention a few of the many names that are familiar to source
level seekers, we have Weston's (1960) benchmark paper that is still
widely used. We have Stockhausen's (1964) data, measurements reported
by Turner and Scrimger (1970), Maples and Thorp (1970), Buck (1974),
and Christian (1965, 1967). Oh, there is no dearth of source level

values. But they do not add up to our desired "known, standard

* With some of the more exotic explosive materials there are problems
with reproducibility; the charge output may vary with charge size,
with the formulation, or with the density. But these materials are
not found among the standard acoustic sources.

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

values." It is true, of course, that the visible problem is not as
great for the lowest frequencies of interest as it is for higher fre-
quencies. Many of the above references do not look at the very low

frequency end of the scale.

SONAR EQUATION COMPATIBILITY AND PREDICTABILITY

In order to examine the problems of sonar equation compatibility
and predictability, we must face up to the nonlinear nature of explosion
pressure waves. Before getting into that discussion, let us take a
brief refresher look at the time and frequency domain functions in this

explosion pressure field.

Figure 1 shows two typical pressure-time histories for underwater
explosions, recorded with the special-purpose equipment designed for
such measurements. The experimental setup is shown at the top of the
figure. Charges were fired at depth and recorded at the surface above.
The record on the left is from a 57-pound TNT charge detonated at
6,600-foot depth. That on the right is from an 8-pound TNT charge at
2,050-foot depth.

These pressure-wave records show the usual high-amplitude shock
wave followed by the succession of pulses associated with the oscilla-
ting bubble of product gases. There are scaling laws for explosion
pressure waves of this sort, in terms of explosive charge material,
weight, and depth of detonation. Unfortunately (from the point of
view of simplicity), these scaling laws contain different coefficients
and exponents for different segments of the pressure wave; and there-

in lies the problem that you will hear more about later.

The frequency domain representation of an underwater explosion

pressure wave, such as Figure 2, is also familiar. But in our everyday

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

treatment of explosion data, where we usually study only narrow-band
slices of the spectrum taken near selected frequencies, we may forget
the oscillatory character of the spectral energy distribution. We
find the highest spectral energy at the so-called "bubble fundamental
frequency," which corresponds to the first bubble period spacing in
the time function. We find minor irregularities in the spacings of
the first several harmonics while the second and third bubble pulses
are getting shaken out. And after a while, the pattern stabilizes,
and we have a regularly spaced series of peaks and nulls. I must
point out that true nulls are in the spectrum, even though they are
not conspicuous at the higher frequencies in Figure 2. This isa
computer-generated plot. The computational grid size used and the
characteristics of the plotter control details of the picture. Lest
you hope to find these oscillations smoothing out at long ranges,

or start thinking of clever ways to clean up the source level curve
by smoothing or filtering (e.g., Skretting and Leroy, 1971), let me
show you one of Gordon Hamilton's sonagrams of signals recorded some
500 miles from explosions (Figure 3). The horizontal axis is time
and the vertical axis is frequency (0 to 500 Hz). Those alternating
light and dark horizontal bands corresponding to no-energy and high-
energy show that neither distance nor manipulation can smooth out the

true source spectrum.

Naturally we are going to have problems in fitting explosion
source levels into the sonar equation, where quantities are added
and subtracted in a comfortable linear fashion. The translation of
"compatibility with the sonar equation" into more explicit prosaic
terms is summarized in Figure 4. What we mean is that we wish there
were no finite amplitude effects; no nonlinearities; no inherent
change in wave shape or frequency distribution as the wave propagates

outward, so that all observed changes could be ascribed to the medium.

Si,

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

The customary method of getting around the finite amplitude
problem is indicated at the bottom of Figure 4. Just go a sufficient
distance from the charge so that the nonlinear effects from that
point on are negligible for the application of interest, call this a

"reference range,"

and there define an effective source level which
can then be used to examine the signals measured at greater ranges.
This practice was initiated by Weston, who chose a 100-yard refer-
ence range. And for the fairly small charge weights and detonation
depths that Weston was treating, most of the finite amplitude effects
are, indeed, negligibly small beyond 100 yards range. Unfortunately,

such quantities have a way of becoming gospel and being dissociated

from the physical facts that led to their selection.

The 100-yard reference range by now has become a sort of junc-

tion through which source levels are shuttled at a rate of 20 log R.

If you want to compare different measurements made at various ranges —

a half mile, a mile or so — assume spherical spreading and extrapo-
late them to 100 yards. If you are enamored of the sonar equation's
one-yard reference range, just add 40 dB to the source level. We
rarely find experiments with data taken on a closely scaled grid
that allow us to see the rate at which finite amplitude effects are
varying, and to sort out all effects of the medium. So we often are
trapped in the circuit of using our desired information to reduce
available data to try to improve our desired information. We must
break out of this circuit if we really want to know effective source
levels within 1 dB for an assortment of charge weights, depths, and

frequency bands.

The fact is that the appropriate reference range for defining
effective source level is itself a function of the charge weight, the

charge depth, and the frequency band of interest. I know of no

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

theoretical treatment that takes as input an explosive charge
composition, weight, and depth and gives in return realistic, reli-
able pressure-time histories at any desired range. This problem has
been worked on for many years, and we are coming closer than Kirkwood
and Bethe might have hoped for. But we are still in an empirical

world when trying to examine source levels parametrically.

Let me show you a sample of data that illustrates the kinds of
wave-form changes we have to deal with. In Figure 5 some recorded
pressure waves are sketched on the right, and the experimental
arrangements used to record them is shown on the left. We lowered
a small oscilloscope housed in a 30-inch diameter sphere down to
about 14,000 feet. A tourmaline gage was suspended below the sphere,
and 50-pound pentolite spheres were suspended some 200 to 1,000 feet
below the gage. We also had pressure sensors near the surface above

the charges.

In the pressure waves shown on the right of Figure 5, solid
lines represent the data measured with the deep oscilloscope near
the charge, and dashed lines represent the same pressure wave
measured near the surface. The upper pair of curves are for a charge
fired 190 feet below the deep gage; the lower pair are for a charge
935 feet below the gage. In both cases, the pressure waves for the
deep (near-field) and shallow (far-field) recordings are plotted on
scales in the same ratio as the stand-off ranges. In other words,
had the waves propagated without changes of shape — simply decreased
in amplitude at the acoustic rate of 1/R — the solid and dashed
curves would coincide. The top set of curves, where measurements are
compared for ranges of 190 and 13,690 feet, shows the well known
shock-wave "finite amplitude effects" of a spreading profile and a
peak pressure that decays more rapidly than 1/R; it also shows that

the same nonlinear behavior is followed in the first bubble pulse.

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

In the lower set of curves, where the close-in measurement was at a
935-foot standoff, the near-field and far-field curves are nearly
coincident. This might suggest that perhaps 300 yards is a reasonable

"reference range" for a 50-pound charge at 15,000 foot depth.

The quality of these few exploratory data is too poor, and the
extent of them too limited, to give the kind of information on source
level that we need. But they do show that we must re-examine the
subject of reference ranges. This is not an academic question. I
am convinced that the methods of extrapolating and interpolating
among different sets of measurements contributes some of the vari-
ation found in reported source levels. And that the choice of
reference range is important to the third and last of our desirable

source attributes: predictability to within 1 dB in 1/3-octave bands.

FACTORS IN SOURCE LEVEL DETERMINATION

As noted earlier, we must still look primarily to empirical,
rather than theoretical, methods of determining source levels. Even
if we had an infinite, homogeneous ocean in which to work, source
level determination would not be easy, because so many factors enter
into the acquisition of the right number. The following list shows

the major factors in more-or-less decreasing order of importance:
@ CHARGE CHARACTERISTICS
- Weight
= Depth
- Explosive Composition
- Configuration
e@ MEASUREMENT RANGE (discussed above)

@ RECORDING EQUIPMENT CHARACTERISTICS

@ PROCESSING METHODS

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

Let me work up from the bottom of the list and give a few examples of
the kinds of variation we have found influencing that desired 1 dB

predictability in 1/3-octave bands.

Processing Methods. The most obvious source of differences from
processing methods is, of course, the fact that some people use analog
and others use digital processing. Even with digital processing, how-
ever, we have been surprised to find how easily some tenths of dB
differences crop up. We recently made a joint study with another
laboratory of selected data tapes — some we had recorded and some
they had recorded. After overcoming the numerous communication
problems involved in exploring our two "Standard FFT programs" (and
this took no small effort), we still found several computational
details that introduced greater differences than one would expect.

For example, simply changing the frequency interval of computation

by a small amount introduced as much as 0.8 dB difference in the
1/3-octave band centered at 25 Hz on some of the records, but not on
others. At 25 Hz, the 1/3-octave band is so narrow that the level

is quite sensitive to small computational manipulations. This
exercise reminded us again of how wary one must be of applying routine
analyses without carefully examining their suitability for a par-

ticular spectral energy distribution.

Recording Equipment Characteristics. The importance of this
factor is so well known that I will mention only one point that may
be of interest. (We are speaking only of equipment that is fully
calibrated, of course.) My example has to do with equipment over-
load. Overloading is familiar to underwater acousticians using
explosion sources. One would like to think that in an overloaded
recording the low-frequency content is still usable and that the
signal has simply suffered high-frequency clipping. Figure 6 shows

an analytical quick-look at this question, using an idealized

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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pressure history for a 1.8-pound TNT charge at 800-foot depth. This
curve (1) was "clipped" at 50 percent of the peak value (curve [21
and again at 10 percent of the peak value (curve [3]) by the computer.
We assumed the most amiable system imaginable, and calculated the
spectrum levels in 1/3-octave bands for the two "clipped" records.

The resultant decreases in levels at the low-frequency end of the
spectrum are tabulated at the bottom of the figure.* When clipped

at 50 percent of the peak — a matter of a few dB — the resultant
error had reached 1 dB in the 250-Hz band. With a 10 percent of peak
clipping, the error was twice our desired 1 dB even down at 35 Hz.
Perhaps down at even lower frequency — one HZ or so — the 10 percent
clipping would not matter. But I think clipped records have to be
handled cautiously. And remember that the possible distortions im-
posed by a system without instantaneous recovery are not included in

this example.

Charge Configuration. The variations in explosion pressure
fields that we can achieve simply by distributing our explosive mate-
rial in different configurations is a complex subject that I will not
even try to touch on today. This discussion is limited to compact,
consolidated, "point" charges that are omnidirectional. But if you
want to modify your spectral energy distribution with a given weight
of explosive, the quickest way is through charge configuration, and

we know a fair amount about the subject.

Explosive Composition. We can accomplish some redistribution of
spectral energy through choice of explosive composition. The bubble
fundamental frequency is a function of the charge weight, the charge

depth, and a material constant. The constants do not differ

* The positive values in parentheses for 25 Hz are spurious and mirror
our failure to do a DC leveling when we clipped the wave.

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

appreciably for typical high-explosive material such as TNT, pentolite,
and RDX. But they are changed significantly when aluminum is added to
the explosive mixture. For example, HBX-3, which is a particular
mixture of RDX, TNT, and aluminum, has a bubble period that is about
25 percent greater than the bubble period of TNT. Consequently, the
bubble fundamental frequency, and the frequency spacings of subsequent
peaks and nulls in the spectrum, are only about 80 percent as great as
those of TNT. This is illustrated in Figure 7, where dashed and solid
lines refer to HBX-3 and TNT, respectively. These curves are computed
from simplified analytical representation of the pressure waves. The
differences in 1/3-octave band spectrum levels of these two materials
are tabulated on the right-hand side of the figure. At the lowest
frequencies, while the curves are still increasing toward the first
peak at the fundamental, the band level for HBX-3 is several dB higher
than that of TNT. As one moves up in frequency, however, it becomes

a game of catch-can, and which of the two materials has the higher
energy level depends on the location of the particular frequency band.
In any case, we expect serious trouble if we try to compare narrow-
band data from two such different materials without accounting for

their different source spectra.

Charge Weight and Depth. I would like to discuss these two
important quantities together for a moment, to describe a method of
source level prediction used by Gaspin and Shuler (1971). The tech-
nique involves first generating a quasi-theoretical pressure-time
history, such as that shown in Figure 8, and then transforming to the
frequency domain. The pressure-time curve is fitted through a series
of points (indicated in Figure 8), the coordinates of which are de-
rived from empirical functions as shown in Figure 9 (Slifko, 1967).
The curve of Figure 9 allows one to estimate the amplitude of the

first bubble pulse, P for selected values of charge depth, Zo!

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charge weight, W, and stand-off range, R. The Gaspin and Shuler model
as it stands is a rather crude one and we need new data to remove some
of its limitations. Nevertheless, in my opinion, the model provides
the best source level estimates that are available, and it has been

widely used.

Varying Charge Depth. In many experiments it is easier to control
the weight of the explosive charge than to control its depth. This is
especially true for acoustic research experiments, where free-fall
charges are often dropped from moving ships or planes, and detonation
depths depend on hydrostatic pressure devices or lengths of fuze. The
SUS Mk-61 sound signal is such a charge. In Figure 10 are shown the
Gaspin and Shuler (1971) theoretical spectra for 1.8 pounds of TNT —
the Mk-61 loading — detonated at three depths, 700, 800, and 900
feet. These three curves indicate the range of source levels one
might encounter with the Mk-61 SUS set at a nominal 800-foot burst
depth. Although detonation depths might vary by only a few feet for
charges drawn from the same lot, the MILSPEC standards are so written
that mechanisms with variations of almost + 100 feet about the nominal
800-foot depth might come within acceptable limits and be included in
stock. The alternate shaded and unshaded frequency bands are the
popular 1/3-octave bands. What these kinds of depth variations mean
in terms of source level uncertainty is shown in Figure 1]l. The
ordinate of Figure 11, AE, shows the dB error introduced when actual
detonation depth deviates from the ideal 800-foot value. At high
frequencies, where the 1/3-octave band encompasses a number of bubble
harmonics, the errors fall within 1 dB of the norm. But down at low
frequencies, where the measurement band width is narrow relative to
the spectrum oscillation pattern, the errors are + 3 dB. So much for
the hope of predicting source levels to within 1 dB, unless the actual
detonation depth is taken into account. Recent efforts along this

line will be discussed by Dr. Weinstein.

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Since I have gone to some lengths to emphasize the need for more
and better data before we can solve source level problems, let me also
mention a case where we will have to rely on processing techniques,
even if we obtain perfect data. This is the shallow burst, where it
is not possible to record the total output wave of the charge separate
from the rarefaction wave that is reflected back into the water from
the surface. For example, the first bubble period of the very popular
Mk-61 SUS fired at 60 feet is about 120 milliseconds, and the entire
train of explosion pulses that comprise the charge output lasts for
several hundred milliseconds. There is no point in the water at
which this pressure wave can be recorded faithfully, because the
longest time interval between the direct and surface reflected waves
that one can find is 24 milliseconds. This maximum interval occurs
directly below the charge, as shown in Figure 12. The parabolic
curves of Figure 12 are isopleths of the time of arrival of the re-
flected wave; values decrease rapidly as the gage location approaches
the surface. (The cognomen "surface cut-off time, ee of Figure 12
is the explosions research community's jargon for the time separation
between the shock front and the surface-reflected wave.) Whether one
unscrambles the two signals by deconvolution in the frequency domain,
as suggested by Hovem (1970) and by Hanna and Parkins (1974), or by
extrapolating time domain functions, as done by Gaspin and Shuler
(1971), some sort of special processing must be applied to obtain
source levels. Figure 13 illustrates the degree of spectrum dis-
tortion introduced by the reflected pulse if it arrives well beyond
the direct wave (top pair of curves), hard on the heels of the direct

wave (center pair), or in the midst of the direct wave (bottom pair).

I have now completed my long list of reasons why it will always
be difficult — if, indeed, possible — to predict explosion source

levels in 1/3-octave bands to within 1 dB at low frequencies. Many

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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ILLUSTRATION OF SPECTRUM DISTORTION

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

of the problems can be removed if we want to pay the price: if we
want to set stringent limits on SUS charge detonation depth toler-
ances, for example, or require high quality on-site measurement of
charge output for each shot in a series. But such prices are high.

I would like to suggest that there may be a better way out, a way that
will involve some possibly painful changes over the short term, but
will pay off handsomely over the long term. Why not work toward a
sensible matching of explosion sources and processing methods? Does

a SUS charge always have to be matched to 1/3-octave bands?

An explosion has an "inherent bandwidth," as it were, in its
bubble fundamental. If analysis bandwidths were selected to be at
least two or three times as wide as the bubble harmonics, much of the
variability I have been discussing will be washed out. On the other
hand, if the practical acoustics problem being attacked includes an
important fixed recording bandwidth which is controlling, then per-
haps we should design a charge to match the problem, not just pick
the one that comes easily to hand. In short, I am suggesting we

shculd try Figure 14.

To summarize: if explosion source levels in 1/3-octave (or
narrower) bands must be predicted to within 1 dB over the frequency
range of 10 to 300 Hz, then:

1) Our present state of knowledge is not adequate.

2) We must acquire new data with controlled experiments.

3) We must improve our source level models.

4) Recording and processing methods must be re-examined.

5) Shallow sources need special attention.

THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

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At the same time that we are working toward improving source
level values as they are used today, we should also be considering
the possibility of a brave new world in which explosion sources and

analysis methods are matched for particular applications.

REFERENCES

Buck, Bellin, ve Acoust. Soc. Ams, 557188-h90 Gi), Lov74.

Chreusizeran, Be An, ue. Acoust. Soc. Am., 42:905-907(L), 1967.

CGheusitian, Be AL, and M. Blak, J. Acoust. ‘Soc. Ams, 38:35/-62, 1965.

Gaspin, J. R., and V. K. Shuler, "Source levels of shallow underwater
explosions," Naval Ordnance Laboratory Report No. 71-160, Oct.

As 7s

Hanna, J. S., and B. E. Parkins, J. Acoust. Soc. Am., 56:378-386,
1974.

Hovem, J: M., J. Acoust. Soc. Am., 47:281-284, 1970.

Maples; lia G-,nand We Hs. horp, J. Acoust. Soe. Ams, 47 (part):
91(A), 1970.

Skeetting, Aj, andi. CG. Leroy, J. Acoust. Soc. Am., 49 (part 2)):
276-282, 1971.

Slifko, J. P., "Pressure pulse characteristics of deep explosions as
functions of depth and range," Naval Ordnance Laboratory Report
Now 6/=S7, Sept. 1967.

Steckhausen, J. H.,;, J. Acoust. Soc. Am., 36:31220(L), 1964.

Turner, R. Ga, and d. A: Scrimger, J: Acoust. Soc. Am, 48 (part 2):
775-778(L), 1970.

Weston, Ds Haj) Proc. Phys. Soc. (London), 77:233-249), 1960:

bys)

CHRISTIAN: THE ACOUSTIC OUTPUT OF EXPLOSIVE CHARGES

DISCUSSION

Dr. J. B. Hersey (Office of Naval Research): We have reviewed
the way we make measurements and have had a very exciting discussion
of what we do to ourselves or for ourselves when we use explosive

charges.

In recent advanced development work in the United States, we
have arrived at the same point that anyone does who tries to make
practical, repeated, continuing application of methods which have
evolved in an experimental atmosphere. We have no choice but to
confirm our measurements in some way. We have to be sure. If the
quantity is a varying quantity, we have to know why it is varying and
how it is going to vary during the life of the application that we
are addressing. This need frequently persists over many years. We,
therefore, have to establish standardization procedures for the
reliability of the techniques that we use. Dr. Marvin Weinstein
has led a rather large group of people in taking a close look at

some of these standardization problems.

EXPLOSIVE SOUND-SOURCE STANDARDS

M. S. Weinstein
Underwater Systems, Inc.
Silver Spring, Maryland

The desired accuracy of low-frequency transmission-loss
calculations based on experimental data using explosive
sound sources is plus or minus one decibel. Factors in-
volved in determining transmission loss include uncer-
tainties in source level, background noise, and process-
ing procedures. Data are presented to illustrate the
quantitative effect of these factors.

While the desired accuracy has not yet been universally
achieved, errors may be minimized through the use of
certain standards concerning source-level monitoring and
data-processing procedures.

INTRODUCTION

The factors involved in propagation-loss determinations using
explosive sources include:
® The desired accuracy
@ The achievable accuracy, stressing the uncertainty
in the source level, the effect of background noise,
and processing procedures
e The information which should be included in technical

reports to permit comparison of data obtained and
published by different organizations.

DESIRED ACCURACY

The first point to consider is the desired accuracy. Our goal
is about one decibel. This statement usually generates immediate

concern, since it has yet to be achieved. There may be basic

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

problems, some of which have been touched on (Christian, these
Proceedings). However, we shall examine some of the concepts which
support 1 dB as a reasonable goal against which to gauge our

performance.

Suppose we are considering the design of a surveillance system
which will make detections at long range where the propagation loss
follows cylindrical spreading. An uncertainty of + 1 dB translates
into an uncertainty in area coverage of about 50 percent; a not

inconsiderable factor in estimating costs.

Suppose we wish to optimize the geographical location or depth
of such a system. One might perform an experiment for simultaneous
measurement of propagation loss at a number of sensor locations. We
want to know the propagation loss difference to within one decibel

for the same reason.

In a somewhat different context, fluctuations in propagation
loss are of considerable interest. If true fluctuations have a
standard deviation of about 3 dB, one cannot stand an uncertainty of

more than about 1 dB before the results are significantly degraded.

Thus, the data needs indicate the desirability of obtaining

propagation-loss data which are accurate to about one decibel.

SOURCE-LEVEL UNCERTAINTY

Consider the accuracies which can be achieved. First, the

uncertainty in source level.

Figure 1 shows source levels in 1/3-octave bands for 1.8-pound
charges detonated at 60 and 300 feet. These three data sets illus-

trate the range of values with which we are confronted. The spread

WEINSTEIN:

EXPLOSIVE SOUND~SOURCE STANDARDS

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SOURCE-LEVEL COMPARISONS

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

in source-level estimates is as high as 7.7 dB. The three data sets

also have different spectral shapes.

The first data set consists of the detailed computation of
Gaspin and Shuler (1971). Although based on experimental data, it
does involve the extrapolation of the shock-wave impulse from
measurements at deeper detonation depths. These extrapolated values

are a good bit higher than those previously used.

The second data set is provided by Busch (1973) and corresponds
to experimental measurements using MILS hydrophones. The system
sensitivity is computed, and not measured. Additionally, correc-
tions for surface reflections are made during processing, since
the reflected signals arrive before the direct signal of shock wave

and bubble pulses has died down.

The third data set is computed using the simplistic forms
published by Weston (1960), which were based on the earlier experi-

mental work of Arons and Yennie (1948).

Although the uncertainty in source level represented by the
spread in these data sets is many times larger than our goal of
+ 1 dB, the problem is recognized and it is reasonable to assume
that the uncertainty in source level can be significantly reduced

by performing carefully controlled experiments.

SOURCE-LEVEL FLUCTUATIONS

When measured data are compared to model computations, we
would like the data to be free from random variations. To help
satisfy this need, the source conditions should be identical from
shot to shot. Fluctuation artifacts can arise from variations in

shot depth which alter the spectral shape, principally by changing

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

the bubble-pulse period, and by variations in yield which change the

total energy output.

Additionally, depth variation will alter the multipath propaga-
tion structure, introducing a third source of fluctuations which will

be dependent on the environmental details.

During a recent experiment, SUS shots were monitored with a
hull-mounted transducer on the drop vessel. The data were processed
to determine the bubble-pulse period for approximately 700 shots
each at 60- and 300-foot detonation depths.

Figure 2 shows the cumulative distribution for the 60-foot shots;
the bubble-pulse period at the top, the detonation depth at the
bottom, assuming that the yield was identical for all shots. The
variation in yield expected from an examination of manufacturing
procedures results in an uncertainty in detonation depth of about
+ 2 feet. As is apparent, the detonation depth varies from 50 to
70 feet. About 90 percent of the data points lie between 54 and 60
feet. For this range of detonation depths the source-level variation

in the 1/3-octave bands is about + 1 dB.

Figure 3 shows similar results for the 300-foot shots with a
data spread of 250 to 350 feet. About 90 percent of the data points
lie between 270 and 320 feet, and source-level variations of about

+ 1.5 dB can be expected.

For both detonation depths source-level variations directly
attributable to variation in yield are estimated to be a fraction

of a decibel.

These results suggest that source-level variations can be

corrected to within a fraction of a decibel by monitoring the shots

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

Bubble-Pulse Period (ms)

110 120 130 140

99 99.

Percent
~
oO

Figure 2.

70 66 62 58 54 50
Depth (ft)

CUMULATIVE DISTRIBUTION OF BUBBLE-PULSE
PERIOD AND DERIVED SHOT DEPTH FROM THE
MK 61 (60 FEET) SUS CHARGES DROPPED BY
THE USNS SILAS BENT. 654 SHOTS.

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

WIRE)

Percent

Bubble-Pulse Period (ms)
38 40 42 44 46 48 50

350 330 310 290 270 25:0;

Depth (ft)

CUMULATIVE DISTRIBUTION OF BUBBLE-PULSE
PERIOD AND DERIVED SHOT DEPTH FROM THE
MK 82 (300 FEET) SUS CHARGES DROPPED BY
THE USNS SILAS BENT. 655 SHOTS.

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

to determine the bubble-pulse period, provided we have either an
experimentally determined data base, or a proven computational
procedure, which permits us to relate depth variation to source-

level variation in the processing bands of interest.

EFFECT OF NOISE

The second source of uncertainty is background noise. To deter-
mine propagation loss, one integrates the signal and noise over the
signal arrival interval and subtracts out the noise from an estimate
obtained by measurement prior to signal arrival. The time interval
between the measurement of noise, and signal plus noise, is typically
about one-half minute. If the noise varies over this time interval,
an uncertainty will be introduced in the computed propagation loss.
Ordinarily, by using only data with high signal-to-noise (S/N)
ratios, perhaps 10 dB or more, this problem is minimal. However,
in large-scale experiments employing many ships, thousands of shots,
and automated remote recording systems, the problem may be more
significant. If the S/N level is less than desired, quality assurance
techniques must be applied to extract the good data and reject the

poor data.

In Figure 4, the error in propagation loss is plotted as a
function of S/N ratio for changes in noise level of + 0.2, + 0.4 or
+ 1.0 dB. As one would expect, the error increases as the signal-to-

noise ratio decreases.

This problem was encountered in a recent experiment. The
following figures illustrate the staged improvement in the quality

of data as quality assurance procedures were applied.

S/N (pB)

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

PL Error (pB)

Figure 4. PROPAGATION LOSS ERROR AS A FUNCTION
OF S/N FOR INDICATED CHANGES IN
NOISE LEVEL

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

The central plot of Figure 5 shows the propagation-loss data.
The ambient-noise levels which were received concurrent with the
shots are plotted in the upper part of Figure 5 at the ranges of
the shot. The resulting S/N ratio for each data point is shown in
the lower set of data. S/N ratios as low as -10 dB had been
accepted at this stage. Note that the background noise shows a
large increase and that the propagation loss follows this change.
Also note that the S/N ratio is poor over this entire region. What
is happening is simply that following the noise measurement a noise
burst coincidentally occurs at the approximate time that the SUS
signal was expected, which is read as signal plus noise, so that,
in effect, a noise fluctuation is mistaken for signal and an

erroneous propagation loss is computed.

It is important to note that the processing system was fully
automated. Ina large experiment, the product of the number of hydro-
phones, shots and frequency bands of interest is of the order of too
Automation is essential to handle this quantity of data. The care
and subjective experience which the scientist can apply when process-
ing data by hand have to be converted to definitive algorithms for
the computer to make a decision. This is not an easy task, particu-
larly for those qualitative factors which the scientist does not

verbalize but applies by gut feeling. If a 10-dB S/N ratio require-

ment were applied to this data set, nothing would be left.

Based on the preceding curves, we therefore decided to reject
all data for which S/N was less than -3 dB, and plotted the remaining
data with different symbols for S/N of -3 to O GB, O to +3 dB, and
greater than +3 dB. The result is shown in Figure 6. Data rejected

for poor S/N are shown along the bottom at the appropriate range.

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These data have been cleansed considerably, but some artifacts
are still present. The total number of data points which remain is

considerably reduced so that other criteria can be applied.

It is known that the nulls and peaks in the source spectrum,
resulting from the bubble pulse, are retained in the spectrum of
signals at long range. Fortunately, narrow-band FFT processing was
a part of the automated processing procedure, so that the spectra
could be examined. The criteria applied was go/no-go. If the
signal spectrum looked like a shot, the data point was accepted;

if it did not, it was rejected.

Figure 7 shows the circled points that were rejected on the
basis of the spectral criteria. The remaining data can now be
relied upon. Further investigations of noise fluctuations permitted
the establishment of estimated uncertainty bars in signal-to-noise
bins. These exceeded our goal of +1 dB, exclusive of the uncertainty

in source level.

Figure 8 shows the spectrum for signal plus noise on the left,
and the noise alone at the right for a 300-foot shot at a high S/N
ratio. Note that the signal plus noise shows pronounced scalloping
with strong nulls spaced at about 25 Hz, consistent with the source

spectrum expectation. The noise spectrum is totally different.

Figure 9 shows similar results for a lower S/N ratio. The

signal-plus-noise spectrum is still good.

Figure 10 shows the results for a contaminated sample. Note
that the signal-plus-noise spectrum does not show the null sequence,
and is quite similar to the noise spectrum. This is a case where
signal plus noise is dominated by a noise burst and this data point

is therefore rejected.

EXPLOSIVE SOUND-SOURCE STANDARDS

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WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

PROCESSING PROCEDURES

Another source of uncertainty results from processing procedures.

This is a broad subject in itself, which we touch upon only briefly.

Any processing system has an inherent limit on reproducibility
which may stem from the tape-playback system, the processing equip-
ment, or human factors. To illustrate the effect on propagation-
loss measurements, assume a repeatability of + 0.2 dB, uncorrelated

for both the noise energy and the signal-plus-noise energy.

The uncertainty in propagation loss, as a function of the S/N
ratio, is shown in Figure ll. For a signal-to-noise ratio of - 3 dB,
the uncertainty is then + 1.0 to - 1.3 dB, about equal to the total
accuracy goal. Also note that because of the asymmetry, a small

fixed bias can be expected.

The results of a repeatability experiment using five 10-second
noise samples from a direct-record ACODAC system are shown at the
top of Figure 12. The operator exercised considerable care in tape
handling to ensure proper lay up of the tape and drive-speed
stabilization. The differences are plotted for 1/3-octave bands.
Analog filters were used. The results range from + 1.0 to - 1.2 dB,
with zero bias and a standard deviation of 0.36 dB. The lower curve
shows similar results without care. Fast forward and reverse were
used to find the data segment of interest. The results are con-
siderably worse over the entire band, and get completely out of hand
at the higher frequencies. This particular data set typifies the

human-factors problem when direct record is used.

Repeatability measurements with an FFT processing system for
the 50-Hz band yielded similar results; a bias shift of about 0.25 dB

and a standard deviation of about 0.25 dB. However, the comparison

Wel

EXPLOSIVE SOUND-SOURCE STANDARDS

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Figure ll.

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

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20
25
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160
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WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

of propagation loss with good S/N ratio was surprisingly good;

essentially no bias and a standard deviation of 0.14 dB.

The automated systems used to process a large quantity of data
are computer controlled but can employ either analog filtering or FFT
processing. In the latter case, the FFT spectral levels are summed
in the time domain to cover the total signal interval, and are
summed in the frequency domain to construct square 1/3-octave or
one-octave bands. A comparison of analog and FFT processing for
noise, with everything else the same, yielded a bias of about 0.3 dB
and a standard deviation of 0.8 dB in the 50-Hz 1/3-octave band.

For propagation loss, again with good S/N ratio, there was essentially

zero bias, and a standard deviation of about 0.5 dB.

Christian (these Proceedings) has pointed out that in processing
one set of short-range recordings, the level changed by as much as
0.8 dB as the FFT bandwidth was changed. This observation is unex-
plained. There are a number of difficulties when we seek high
accuracy. Concerning FFT processing, it appears that most of our
knowledge is derived from consideration of long-duration Gaussian
signals. Explosive signals do not satisfy this criterion. They
consist of a series of short transients with deterministic spectral
characteristics. Specifically, we must know how the broadband FFT

levels depend upon:

e The bandwidth selected for processing
e The digitization rate

e The number of bits

e Whether Hanning is or is not used

e Whether coherent or incoherent summation in the
time domain should be used.

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

It is illustrative to compare data processing results of dif-
ferent facilities. Each facility used a different duplicate of the
original recording, either direct or FM, depending upon available
equipment. Figure 13 compares about 100 shots for Systems (1) and
(2). System (1) is fully digitized using FFT processing. System (2)
uses analog 1/3-octave filters followed by digital processing. The
comparison is for the 50-Hz band only, for three different hydro-
phones. Independent processing yielded biases from + 1.47 dB to
+ 2.13 dB, and standard deviations from 1.28 dB to 2.06 dB. One of
the problems identified at this time was that the two facilities
made independent and different determinations of signal duration.
When these were made consistent and the data reprocessed, the bias
was reduced to between + 0.95 and + 1.85 dB, and the standard

deviation to between 1.05 and 1.52 dB.

To help identify the reasons for the observed differences, the
tape recording used on System (2) was then processed with System (1).
The additional improvement was a few tenths of a decibel, identify-
ing the major source of the differences shown as resulting from

differences in the processing systems.

A comparison between System (1) and another hybrid analog-
digital system (System (3)) is shown in Figure 14. The agreement
is somewhat better. These results, combined with the prior compari-
son of analog and FFT processing, with all else the same, suggest
that the differences seen in the previous figure are not totally

attributable to differences between analog and digital processing.

Figure 15 compares System (1) and a totally analog shot
processor, System (4). It employs 1/4-octave rather than 1/3-octave

filters. A simple 10-log bandwidth ratio correction was made.

EXPLOSIVE SOUND-SOURCE STANDARDS

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WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

Summarizing the last three figures, we have propagation-loss
uncertainties consisting of both biases and standard deviations as
high as about 1.5 dB, which is not a particularly satisfying accuracy,

especially since it does not include the uncertainty in source level.

DATA COMPARABILITY

Based on the investigations which have been conducted, it is
clear that the source levels and processing bandwidths should be
documented to permit data comparisons. Some indication of the
quality assurance procedures used, or a best estimate of data accu-

racy is useful.

Ultimately, it appears desirable to develop standardized pro-
cedures for checking or adjusting processing systems. Some thought
is currently being given to this in the LRAPP community using
recorded transients to simulate explosive signals but constructed
so that the spectral content can be computed by closed analytic

forms.

REFERENCES

Arons, A. B. and D. R. Yennie, "Energy Partition in Underwater
Explosion Phenomena," Rev. Mod. Phys. 20, 519, 1948.

Busch, J. M., "Spectra of Explosive Sound Sources: Mark 82 - Mod 0,
Mark 64 - Mod O, and No. 8 Vibro-Cap," Bell Laboratories, OSTP-12,
November 12, 1973.

Gaspin, J. B. and V. K. Shuler, "Source Levels of Shallow Underwater
Explosions," Naval Ordnance Laboratory, NOLTR 71-160,
November 13, 1971.

Weston, D.E., "Underwater Explosions as Acoustic Sources," Proc.
Phys. Soc. (London) 76, 233, 1960.

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

DISCUSSION

Dr. D. V. Wyllie (Weapons Research Establishment (WRE), Adelaide,
South Australia): We are very interested in propagation-loss measure-
ments using explosive charges as sound sources. We are also concerned
with the difficulty in assigning a precise source level to these charges
at frequencies around 20 Hz. Third-octave bands are unsuitable for use
at these frequencies since they are too narrow and the resultant source
level too uncertain. In our measurements we shall be employing octave

bands. However, the source level still remains a problem.

Something we have considered is changing the pressure signature of
the charge through the use of explosives other than TNT. There are

groups within WRE interested in pursuing this approach.

Since the variation in level in narrow bands at low frequencies
results from the interference between the radiation from the pressure
and first-bubble pulse, the use of rocket propellant as the explosive
could mcdify the pressure signature of the charge sufficiently to reduce
level variations at low frequencies. The pressure signature can be
modified by varying the burn rate of the explosive. The resulting
explosive may be more useful at low frequencies. Ms. Christian, are
you aware of previous work along these lines and would you like to comment

on the approach?

Ms. E. A. Christian (Naval Surface Weapons Center): The fact is,
yes, you certainly do have materials which have slower detonation (or
burning) rates than typical HE materials. But the only way I have ever
heard of for really eliminating this ungainly bubble pulse is through
some mechanical method of breaking it up or venting it out. You can
arrange to have a natural venting, as has been done by Woods Hole for
many years; that is, you can fire the charge near the surface, shallow

enough so that the bubble blows out and it isn't there to bother you.

WEINSTEIN: EXPLOSIVE SOUND SOURCE STANDARDS

However, when using very shallow charges, I think you have a serious
source-level definition problem because you generate a directional
pressure field that is highly sensitive to the exact firing depth.

You no longer are really using the total charge output, so you no longer
have the kind of source levels we are talking about. The best mechanical
way of eliminating the bubble that I have heard about is a sort of iron
maiden with a Swiss cheese skirt. You simply put the charge down into a

big sphere with holes in it and this breaks up the bubble as it forms.

As far as eliminating the bubble pulse problem by varying your
detonation rate in an explosive composition, I think the best you can do
is redistribute your available energies somewhat; and you still come up
with some sort of oscillation. What you are doing is transforming a
solid mass of material into the same volume of gas in a very short time
however much you slow down the detonation. Willy nilly, the gas is ata
high pressure and temperature, it is going to expand rapidly, and then
it is going to collapse. So I really can't see how you could eliminate

the bubble pulse.

Dr. Wyllie: I wasn't talking about eliminating the bubble pulse,

but rather the pressure pulse.

Ms. Christian: Well, in any case, it seems to me you are going to
have a double pulse wave, even if you effectively cut off the top of

your shock wave and have a slow-rising sinusoidal first pulse.

We have done a little work along those lines, not very much, using
detonating black powder and an ARP propellant. We had problems of
reproducibility with those materials. We found, for example, that
with black powder you must have a very high containment to make the
charge detonate reliably. And with the propellant indeed, you do cut
off the top of the shock, but you still have the bubble. So if the

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

bubble is there oscillating, I would think you are still going to have

an oscillatory spectrum.

Most likely you would cut off the level of the high-frequency end of
your spectrum as you cut off the top of your sharp shock. So you have
lost energy at the high-frequency end. You may not have lost as much
energy at the low-frequency end as at the high, but possibly some is
lost there as well. As I see it through my cloudy glasses, the problem
is that so long as you have an oscillating bubble you don't have a white-
noise spectrum, and I don't see how you can remedy this with chemical

composition alone.

Dr. G. B. Morris (Scripps Institute of Oceanography): I believe
the oil industry has had this problem of bubble pulses for several years
and has in the past few years effected quite a number of solutions.

One such system uses injections of high pressure steam, such that the
steam condenses into water, eliminating the oscillating gaseous bubble.
Other systems make use of what is known as a "Sleeve exploder." A
propane-oxygen mixture is injected into a perforated tube covered by a
rubber sleeve. Upon detonation the sleeve contains the gaseous products
which after full expansion are vented to the surface to prevent the

bubble pulse.

Devices like these might get away from the bubble problem, although
I suspect these devices give a lower energy output. The signal-to-noise
ratios will be lower, and the resulting propagation measurements will be
subjected to the signal-to-noise ratio problems discussed by Dr. Weinstein.
What you have gained at one end by eliminating the bubble, you have lost
at the other end by having a lower energy output source. However, it

might be worthwhile examining the outputs of some of these sources.

WEINSTEIN: EXPLOSIVE SOUND SOURCE STANDARDS

Dr. M. Schulkin (Naval Oceanographic Office): What do you estimate

is the high frequency cutoff for your oscillograms for those charges?

Ms. Christian: About 100 kHz. The response is flat up to about

100 kHz in oscilliscope readings.

Dr. Schulkin: We found that the peak pressure that you actually
measure very much depended on the high-frequency response of the gage.
For many systems used at sea, the tape recorder itself cuts off that

peak.

Ms. Christian: Right. Actually, you can get within the typical
reproducibility of the data points probably if your recordings are good
up to about 20 kHz. You don't lose much in the peak pressure above that
frequency. That is, you really can't see the very high-frequency spike
above about 20 kHz. But if you cut off at, say, 5 kHz, you can be down
30 percent in the peak pressure. So all of those wave functions we
use in the Gaspin and Shuler model require this extremely broad-band

recording to give a true wave form.

Dr. Schulkin: I have seen records published where the peak and the

first-bubble pulse have the same amplitude.

Ms. Christian: Oh sure, you will get to the point where the
bubble peak pressure is higher than the shock if your cutoff frequency
is sufficiently low. As a matter of fact, I think at an upper limit of

about 500 Hz the bubble and shock are equal.

Mr. G. R. Hamilton (Office of Naval Research): What is the uniformity
of our standard SUS from SUS to SUS if you detonate them all at the
same depth? You talked about variations from variation in depth.
What's the variation in source level if we fire them at precisely the

same depth?

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

Dr. Weinstein: I have to honestly answer that we don't know. We
don't have a set of data that would answer the question for us. We

have been discussing means of finding out.

Mr. C. W. Spofford (Office of Naval Research): I have a comment
for Dr. Weinstein. I am concerned with the measurements of transmission
loss at low signal-to-noise ratios, especially when you might have many
low signal-to-noise arrivals adding in the shot processor versus one
larger arrival yielding the same total signal-to-noise ratio. The
question is whether or not the accuracies of these two measurements are
comparable. I think there may be less accuracy in the first measurement

than the second, even when you appear to have 3 or 4 dB of signal-to-noise.

Dr. Weinstein: I think it goes the other way. If you have plus
3 dB signal-to-noise based on the total integration, and if you look at
the peak of the individual arrivals and your multi-arrival structure to

noise, your S/N would be a lot higher.

Mr. Spofford: I guess I'm concerned about losing arrivals down in
the noise even though the final transmission loss appears to have adequate
signal-to-noise. You may have lost the low amplitude arrivals in the

noise.

Dr. Weinstein: The problem is we have noise and multiple arrivals.
The signal-to-noise that I am talking about is what is obtained by doing
a total integration over the multiple arrivals. You will obtain a lot

lower signal-to-noise ratio than you would obtain if you were to define

it based on the peak of one of your multi-path arrivals to noise background.

Dr. J. S. Hanna (Office of Naval Research): The question of
signal-to-noise ratio that I believe Mr. Spofford was getting at is not
the sort of thing that you get from looking at the peaks of those traces.

He is interested in signal-to-noise ratio in a third-octave band around

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

some frequency. The question is what is the signal-to-noise ratio of
each successive arrival? That's not directly inferrable from the broad-

band pressure versue time history you showed.

You may have a very large signal-to-noise ratio based on peak
amplitude but at some particular frequency have a very poor signal-to-
noise ratio for that same arrival. So the point is, you may have some
very marginal signal-to-noise ratios in that band arrival by arrival

and yet have a total energy that is misleading.

Dr. Weinstein: Yes, I understand what you are saying. Why don't
we consider the time series of the signal in a third-octave band
already? If we were to look at a part of it, it would have a much
higher signal-to-noise ratio than we would have for the total. Yes,
there may be some arrival which we are not seeing which has a poor
signal-to-noise ratio. But you have to recognize that we have time domain
problems here. This harks back to the question of overloads in the
system. Can you take an overloaded signal and make some estimate as
to what the propagation loss level has to be at which you will overload?
Well, the answer is you can't because the propagation loss depends

upon the multistructure. The overload depends upon the individual peak.

Dr. S. C. Daubin (Rosenstiel School of Marine and Atmospheric
Science, University of Miami): I want to ask a question related to
Mr. Hamilton's question of variability from shot to shot at the same
depth. Could you tell me what the manufacturing tolerance in a SUS is
regarding the weight of the charge? Is it 1.8 pounds plus or minus

what?

Dr. Weinstein: I don't remember the number precisely, but my
recollection is it's going to be plus or minus a couple of tenths of a
pound, something of that sort. But the problem doesn't lie in that

tolerance. The problem lies in how the SUS is manufactured.

oil

WEINSTEIN: EXPLOSIVE SOUND-SOURCE STANDARDS

There can be as many as three pours for a single SUS with hardening

and, therefore, layering between pours, so you have that additional

problem.

APPLICATION OF RAY THEORY TO
LOW FREQUENCY PROPAGATION

Henry Weinberg

Naval Underwater Systems Center

New London Laboratory

Reprinted from NUSC Technical Report 4867, 18 December 1974.

938

ABSTRACT

The development of ray tracing techniques is reviewed,
and then the effects of various sound-speed representations
on the computed value of propagation loss are discussed.
Since modified ray theories designed to treat caustics lose
their effectiveness at the lower acoustic frequencies, an
alternative approach for the horizontally stratified case
is proposed. For oceans that are nearly horizontally
stratified, the method of horizontal rays is applicable.
Computed predictions are compared with measured data.

APPLICATION OF RAY THEORY TO
LOW FREQUENCY PROPAGATION

INTRODUCTION

It is often said that ray theory is not applicable to low frequency propagation
in the ocean. The purpose of this report is to demonstrate that this is not the
case. If the word "ray" is allowed a more general meaning than that used in the
classical sense, then ray tracing is indeed a useful means of modeling low fre-
quency propagation.

Early ray tracing programs were primarily concerned with integrating the
ray tracing equations of the next section accurately and efficiently. It is shown
that the effect of sound-speed representations on the computed value of propa-
gation loss is not as important as is currently believed. The most recent addi-
tion to practical ray tracing programs is the asymptotic treatment of caustics.

In the case of a horizontally stratified ocean, the integral representation
may be expanded into a multipath series, each term of which corresponds to a
particular ray type. Upon integrating, one obtains the acoustic pressure along
the ray. It is important to note that this multipath expansion is exact. The ac-
curacy of the final result depends on the method of solving the depth dependent
wave equation and evaluating the ray type integrals.

For low frequency propagation in nearly horizontally stratified oceans, the
method of horizontal rays is recommended. Here, the pressure is expressed as
a summation of normal modes weighted by amplitudes satisfying horizontal ray
tracing equations.

RAY TRACING EQUATIONS
Several years ago, the state of the art was described in Officer's! book on

sound transmission. Then, ray tracing involved approximating the solution of
the reduced wave equation for the acoustic pressure P

2 eae
Vv ee) P=0

with the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) form

P =a exp (iwT) .
The travel time T and amplitude a satisfy the eikonal equation

(WOE ec
and the transport equation
Vv: a*VT = 0,
respectively.
The eikonal equation may be solved by using the method of characteristics
for first-order partial differential equations. Characteristic curves, better

known as rays, are orthogonal to surfaces of constant time. They satisfy the
ray tracing equations

% | 2
—~
Qo|F
& | &
So
ll
le
alr

Spot a

Once the rays have been found, the divergence theorem applied to the trans-
port equation produces the geometrical spreading law for the pressure amplitude,

5 fe 50 ie 2

— — ——_— +
c 60,

ag 0

or for the equivalent plane wave intensity,

= |=
Ip es)

The subscript zero refers to a reference point (usually 1 yd away from a
point source), and $0 is the cross-sectional area of an infinitesimal ray tube.
The intensity satisfies the conservation of energy law along a ray tube. Itcan
be shown that pressure, on the other hand, satisfies the law of reciprocity.

In the case of a horizontally stratified ocean, that is, the sound speed and
ocean boundaries are independent of both horizontal coordinates, the rays re-
main in a vertical plane and obey Snell's law,

iar
— — = constant ,
e ds
where
7 Ss rae y2

SOUND SPEED REPRESENTATIONS

When ray theory was first implemented on digital computers, the primary
concern was to integrate the ray tracing equations accurately and efficiently.
Pedersen2 motivated the design of many ray tracing programs by demonstrating
the fact that discontinuities in the sound speed gradient could introduce anomalies
in the computed value of geometrical spreading loss.

This effect is illustrated by fitting the sound speed profiles of figure 1 with
piecewise linear, 1 piecewise quadratic, 2 and cubic spline? representations.
Differences are more readily seen in the sound speed gradients shown in this
figure. The corresponding ray diagrams, figure 2, show that caustics due to
discontinuities in the gradient of the piecewise linear fit disappear when smoother
sound speed representations are employed. Propagation losses computed ac-
cording to classical ray theory tend to accentuate this effect, but consider what
would happen if a ray theory generalized to treat caustics correctly were used
instead. Figure 3 indicates that the effect of different sound speed representa-
tions is insignificant providing that each representation accurately describes the
input data to be fitted.

DEPTH (m)

1496

120

160

200

120

160

200

120

160

200

Figure 1,

VELOCITY (m/sec) GRADIENT (1/sec)
1500 1504 -0.10 0 +0.10
——————

Piecewise Linear

Piecewise Quadratic

Cubic Spline

Comparison of Sound Speed Representations and Gradients

RANGE (km)

Piecewise Linear

Piecewise Quadratic

1201

160+

Cubic Spline

Comparison of Ray Diagrams for an Axial Source

Figure 2,

Piecewise Linear

\ Piecewise Quadratic

PROPAGATION LOSS (dB//1 yd)

Cubic Spline

0 1 2 3 4 5
RANGE (km)

Figure 3. Comparison of 1-kHz Propagation Losses for a 40-m Receiver Depth

100

As far as ray diagrams and propagation losses computed according to clas-
sical ray theory are concerned, past experience indicates that cubic splines
produce the best representations for analytic type sound speeds. However, as
the input data become more irregular, the curve fitting procedure becomes more
difficult to automate.4 A second disadvantage of cubic splines is that the corre-
sponding ray tracing equations cannot be integrated in closed form, whichis a
process that can be accomplished with piecewise linear and quadratic fits.

Many of the statements made above are also true when the sound speed
varies with one or more horizontal coordinates as wellas depth. If, for example,
the input data are fitted with triangular planes, the ray tracing equations may
be integrated in closed form, but anomalies due to discontinuous gradients are
possible.

ASYMPTOTIC TREATMENT OF CAUSTICS

In the last few years, significant advances inpractical ray tracing techniques
involved the treatment of caustics rather than improvements in curve fitting
algorithms. The problem may be illustrated when the sound speed decreases
inversely as the square root of depth, as shown in figure 4. We see that the
ray diagram, figure 5, forms a well defined caustic.

VELOCITY (kyd/sec) GRADIENT (1/sec)
] 2 =2 0) =la2 054 5 O74

DEPTH (kyd)

Le@

Figure 4, Sound Speed and Gradient Studied by
Pedersen and Gordon

101

Figure 5, Ray Diagram for a 1-kyd Source Depth
Computed by Using the Sound Speed of Figure 4

Pedersen and Gordon® compared the classical solution (solid line) with
Brekhovskikh's® modified ray theory (broken line) in figure 6. Classical ray
theory predicts an infinite intensity at the caustic at 3159 yd and an infinite
propagation loss in the shadow zone to the right of the caustic. Pedersen and
Gordon explain that the abrupt change in loss at 3130 yd occurs at the ray that
grazes the ocean surface. The modified ray theory did not apply to the left of
this ray.

The above remark points out the difficulty of applying modified ray theories
to the simplest of caustic geometries. Additional effects due to the ocean bound-
aries, cusped caustics, etc., prevent the theory from being applicable every-
where. One can program as many special cases as practical considerations
suggest, but, more often, one uses classical and modified theories outside
their domain of validity. Since caustic corrections are usually obtained by
including additional terms of a high frequency expansion, errors increase as
the frequency decreases.

102

we 60

os :
a CLASSICAL RAY THEORY ;
° y
Fl

)

a 70

MODIFIED RAY THEORY “#7?!

80 2 : F
3100 3110 3120 3130 3140 3150 3160 3170

RANGE (yd)

Figure 6, Comparison of 2-kHz Propagation Losses for a 0, 8-kyd Receiver
Depth Computed According to Classical and Modified Ray Theories
(After Pedersen and Gordon, reference 5,)

103

(on)
(2)

PROPAGATION LOSS (dB)

NAVA |
70

3.10

3.12 3.14 3.16
RANGE (kyd)

Figure 7. Propagation Loss for a 0. 8-kyd Receiver Depth
Computed More Accurately Than That for Classical or
Modified Ray Theories

Consider what would happen if atheory generalized to treat caustics correctly
were used instead. The result, figure 7, indicates that there is no discontinuity
in propagation loss at the ray that grazes the ocean surface and also that classical
ray theory appears to be more accurate to the right of the grazing ray than to
the left. Consequently, modified ray techniques should be exercised with caution.

Spofford was one of the first to implement modified ray theory in a practical
computer program. The procedure, based on the work of Ludwig, 7 assumes that
the reduced wave equation has an asymptotic solution of the form

2/3 Ai' os

P = exp (iwT)}{ gAi w p) +

subject to the orthogonality condition

VT-Vp=0O0 ,

104

where Ai is the Airy function of the first kind, and T, p, g, andh are to be
found. Upon substituting this ansatz into the reduced wave equation and compar-
ing similar powers of frequency, one obtains

T= (Ee )/ 25,

2/3 p/% = (7, -T_)/2,

g = pi/4 (a. 4 a5)/72 .

h = nae (a, + a_)/2

As illustrated in figure 8, subscripts + and - refer to the two rays that touch
and do not touch the caustic before reaching the field point, respectively. There-
fore, all the quantities appearing in Ludwig's representation may be expressed
in terms of the travel times T; and amplitudes a; of classical ray theory.
Brekhovskikh's solution lacks the term involving the Airy function derivative, a
term that is important away from the caustic. As a result, Ludwig's solution
has a larger domain of validity.

- CAUSTIC
ae

Figure 8. Classical Rays Used to Compute a
Uniform Asymptotic Solution at a Caustic

LOW FREQUENCY PROPAGATION IN HORIZONTALLY
STRATIFIED OCEANS

Most of the figures discussed before were produced by a computer program
designed to model acoustic propagation in a horizontally stratified ocean. For
mediums such as this, the acoustic pressure due to a unit point harmonic source
situated at (0, 0, zg) has the integral representation

foe)
P(r, Z, 2.3) = of wrtJo (AL) G (Z, Zg3A,w) dA,
fe)

where the Green's function G satisfies the depth dependent wave equation

[a?va z2 + w2 { 7? (z) - 7h G(Z, Zg3A,w) = -26 (Z-Zg)
and suitable boundary conditions.

The method of solution used here, that is multipath expansion of the integral
representation, is quite old, dating back nearly 40 years to Van der Pol and
Bremmer. 8 Following Leibiger and Lee,9 the Green's function is expressed
in terms of two linearly independent solutions F; of the homogeneous depth
dependent equation. The solutions F; are normalized so that their Wronskian
equals -2wi. Upon expanding the denominator of G into a geometric series,
the double summation

00 4
P(r) 2, -Zg30) =) D> DY pw (fT, 2, 2,3 w)
v=0 n=1

is obtained. If z< Zs, one sees that

Bee (z; Zy Zg3 w) + iwJ, (wr r) F_(Z; A,w) Fy (253d, w)
oO

Your (A2®) Yop (Are) ar,

where Ygyur and Ypot are boundary reflection coefficients. Other terms of
the series are similar, each integral representing a particular ray type. The
first four are illustrated in figure 9. It is important to note that, so far, the
solution is exact. The validity of the final result depends on the method of

106

RANGE

DEPTH
i

Figure 9. The Principal Ray Types

solving the depth dependent wave equation and evaluating the ray type integrals.
If WKBJ and stationary phase techniques are used, respectively, the classical
ray theoretic solution is obtained. Murphyl0 replaced the WKBJ technique with
a Weber function representation in order to treat the two-turning-point problem.
Brekhovskikh replaced the method of stationary phase with an Airy integral
modification in order to investigate caustics.

At present, our propagation model uses the following modified WKBJ ex-
pression to solve the depth dependent wave equation:

1/2 w 1/6 exp { + iw (2,9 Z 3X) Fin/a}

8(2Z3d) (Bi {2/3 P (Zz; »} ae iaifu”/S p sn) ,

a 1/2
Q (29, asnr=| Jeena ye odie,
Z

Fy (Z3X,wW) = *

re)
3 2/3 a(zsn)| “2/2
P(Z;X) ra -{2 Q (Z, 2430) 9 and foe (Z3 d) = 32 ;

107

Z, is a suitably chosen reference point

z, is a turning point

Bi is the Airy function of the second kind.

Whenever w2/ 35 is a moderate to large negative number, F, reduces to the
usual WKBJ representation F

Fy (23h, 0) = {o-? mee Ns

exp {+ iwQ (Zos zs) :

Since this modified representation is inaccurate in the vicinity of double turning
points, Fy; are arbitrarily set to zero whenever they occur. Hopefully, this

will only affect a small interval of integration and will not introduce significant
errors in the final result. Murphy's technique offers an alternative procedure.

The method of evaluating the ray type integrals is based on the following:
1. Segment the interval of integration into suitably chosen subintervals.
2. Use stationary phase to evaluate subintegrals whenever possible.

3. Integrate the remaining cases numerically.

It was originally though that the numerical integration, although lengthy
when compared with stationary phase, would be invoked so infrequently that its
contribution to the total computer running time would be inconsequential. So far,
this has not been the case. Hopefully, the running time will be reduced eventually
when the integration routine is made more efficient.

Since it is customary to give computer programs names so that they may
be distinguished from others performing similar functions, the program used
herein is called CONGRATS V, where CONGRATS is an acronym for Continuous
Gradient Ray Tracing System. Actually, the completed program will predict the
performance of active and passive sonar systems and is, therefore, more than
just a propagation program. As shown in figures 1 through 3, CONGRATS V has"
the option to invoke several ray tracing procedures. The propagation losses were
obtained by adding the multipath contributions coherently. CONGRATS V also
produces a plot of propagation loss using power addition, in which case the
phases of the individual contributors are neglected.

108

A COMPARISON OF PROPAGATION MODELS

At present, the state of the art of propagation modeling for stratified oceans
may be illustrated by two figures compiled by E. Jensen of NUSC. (See figures
10a and 11a.) Both compare FFP, 1! Fact, 12 RAYMODE 9, 8 and nissm 1114
computer predictions for 50-Hz propagation in the Pacific. The choice of pro-
grams included in the comparison was mainly of convenience, since each is
available at NUSC, New London, and all but FACT were designed there.

Briefly, the Fast Field Program (FFP) utilizes Fast Fourier Transforms
to evaluate the integral representation. The Fast Asymptotic Coherent Trans-
mission Model (FACT) is a constant gradient ray tracing program incorporating
sophisticated low frequency modifications. RAYMODE 9, the latest version of
the series, uses ray theory to determine which intervals dominate the integral
representation, but uses normal modes to compute the acoustic amplitude. The
Navy Interim Surface Ship Model (NISSM) II is a continuous gradient ray tracing
program designed to predict the performance of active sonar systems. All but
FFP have the option to combine multipath contributions incoherently as well as
coherently, and all but FFP use alternative procedures for surface duct propa-
gation.

As a result, the first case (figure 10a), which is dominated by surface duct
propagation, will show the greater variability. FACT is an order of magnitude
faster than NISSM II and RAYMODE 9, while FFP is a good deal slower.

Upon adding CONGRATS V to the comparison (figure 10b) and invoking the
coherent phase option, one sees good agreement with FFP. Had the incoherent
phase option been invoked instead, CONGRATS V would have agreed with the
others.

The second case, figure 1la is dominated by convergence zone propagation.
The agreement is better than before although running times continue to differ by
orders of magnitude.

Upon adding CONGRATS V to this comparison (figure 11b), one obtains
reasonable agreement with FFP. It is unusual to find agreement among models
that are based upon different theories and written by different programmers.
Unfortunately, comparisons are not always this good. Hopefully, all discrep
ancies will soon be eliminated or at least accounted for.

109

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HORIZONTAL RAY THEORY FOR NEARLY HORIZONTALLY
STRATIFIED OCEANS

At long ranges or in shallow water, the effects of horizontal variations in
the sound-speed or bottom characteristics are often large and not readily modeled
by any of the techniques discussed previously. A nearly horizontally stratified
ocean is one in which the horizontal variation is slow.

This rather vague notion is quantized by introducing a small parameter e
and assuming that the properties of the medium depend n the horizontal coor-
dinates X,Y only through the combinations15

X= eX, y= ey.

This being so, let us seek solutions of the reduced wave equation in the form

foe)
P (x, y, 23 e)= exp { 6(x, y)/ie} De ACK Vor Zen
v=0
Each A,, in turn, is assumed to have the form

te
A, @& Ys 4) > 2, al Vx, Y) W(X Ys 2) >»

where the Yk are orthonormal eigenfunctions of the depth dependent wave equa-
tion

aK 2 2
2 +K (X, Y> Z)v_E = AL Lan

OZ
subject to the appropriate boundary conditions.

Upon substituting our ansatz into the reduced wave equation and comparing
similar powers of ie, one finds that the phase function, @, satisfies the hori-
zontally dependent eikonal equation

2
a0 - 2
(—) + (**) = Mi (X,Y) >
Ox OY,
where Ap is one of the eigenvalues, kK? computed above.

This equation, like the ordinary eikonal equation, may be solved by using
ray tracing techniques. Note,however, that all depth dependence is missing.
The pressure depends on depth only through the vertical eigenfunctions. It may
also be shown that the leading amplitude, al”), satisfies the conservation law

114

2
Ap (-.”) 60 = constant

along a horizontal ray tube.

A computer program based onthe consideration above was written to proceed
in two stages. The first determines the eigenvalues and normalized eigenfunc-
tions at each point of a rectangular grid in the horizontal plane. Then, during
the second part, a set of horizontal ray tracing equations is integrated for each
eigenvalue, and the contributions of individual modes are combined to obtain
the total field.

As in ordinary ray programs, only the leading term of the asymptotic ex-
pansion of each mode is found. The expansion then reduces to that derived by
Pierce almost 10 years ago. 16

The program predicted propagation loss along a 1500-nmi track extending
northward from 27° 30'N, 157° 50'W to 52° 30'N, 157° 50'W. Eleven equidistant
velocity-depth profiles obtained from the measured data displayed in figure 12
were entered into the computer program. Note that the SOFAR axis rises from
a depth of 795 m at 27° 30'N to about 50 m at 52° 30'N. Lack of relevant data
prevented the inclusion of any dependence of sound speed or bottom depth upon
longitude.

Figure 13 displays propagation losses from dynamite charges detonated

500 ft below sea level along the track to a 2500-ft receiver depth situated at
27° 30'N. The top graph represents observational data, while the middle graph
shows computed results. The two are superimposed in the bottom graph. The
figure displays an interesting feature. The propagation loss decreases with in-
creasing range beyond 42°N. This decrease may be explained by the fact that
the receiver is only 124 ft away from the SOFAR axis, where the signal is
strongly affected by the amplitude of the few lowest modes, as shownin figure 14.
As the source ship moved north, the source approached the SOFAR axis causing
the amplitude of these modes (figure 15) toincrease to suchan extent that even-
tually the loss due to horizontal spreading was overcome and the total propaga-
tion loss decreased.

The 10, 800-ft receiver depth of figure 16 is well below the turning points
of the first few modes, andso the signal there is dominated by the higher modes.
The amplitudes of these modes are not greatly affected when the source approaches
the SOFAR axis; therefore, for this receiver, cylindrical spreading dominates
the entire track.

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DEPTH (km)

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1572°50"W

6
1.45 1.50 oo -0.1 0.0

VELOCITY (km/sec) EIGENFUNCTION

Figure 14. Sound Speed-Depth Profile at 27° 30'N, 157° 50'W and the
Corresponding First Four Modes for a 31-Hz Frequency

0.1

DEPTH (km)

50° 0'N
157° 50'W
]
7
$)
4
5
6
1.45 1 go) 1.55 =Oel 0.0 0.1
VELOCITY (km/sec) EIGENFUNCTION

Figure 15. Sound Speed-Depth Profile at 50° 0'N, 157° 50'W and the
Corresponding First Four Modes for a 31-Hz Frequency

ALLS)

100
— 110
[-a)
2120

130

100
zx IO
a
3120

130

100
= 110
2120
130

FREQUENCY 31 Hz
SOURCE 500 ft
RECEIVER —__ 10800 ft

25° N 30° N 35° N 40°N 45°N 50° N

TP Rese

So N

60° N

Figure 16. Propagation Loss versus Range for a 10, 800-ft Receiver Depth,

a 500-ft Source Depth, and a 31-Hz Frequency

120

SUMMARY

Contrary to popular belief, ray theory is an accurate and efficient means
of investigating low frequency acoustic propagation in the ocean, Of course, in
this report ray theory has not been used in its classical sense,

Several illustrative examples proved that it is possible to design a single
propagation model that agrees with analytic solutions and measured data, as
well as other computer programs. This effort is more difficult than one may
realize, for once a computer program is tuned to the actual environmental con-
ditions of a real ocean, it may be impossible to input data for which analytic
solutions are known. The apparently simple task of comparing programs is in
reality even more difficult than a comparison with analytic solutions. First,
one must have access to the programs being compared. Second, the programs
must treat the input data similarly. Finally, the programs must treat the out-
put data similarly. For example, how is one to compare coherent phase propa-
gation loss predictions with those of a random phase program ?

Although all the computer models discussed above have been designed within
the last few years, the theories upon which they are based are much older.
Therefore, it is felt that improved computing facilities rather than improved
acoustic theories have been responsible for improved prediction capabilities.

The future of ray theory may prove quite different. Application to unstrat-
ified media, random media, etc. is the next logical step, but these theories need
more development before they can be implemented into practical prediction
models.

P20

10.

11.

12.

REFERENCES

C. B. Officer, Introduction to the Theory of Sound Transmission, McGraw-
Hill Book Company, New York, 1958.

M. A. Pedersen, "Acoustic Intensity Anomalies Introduced by Constant

Velocity Gradients, ' Journal of the Acoustical Society of America, vol. 33,
no. 4, April 1961, pp. 465-474.

C. B. Moler and L. P. Solomon, "Use of Splines and Numerical Integra-

tion in Geometrical Acoustics, " Journal of the Acoustical Society of America,
vol. 48, no. 3, September 1970, pp. 739-744.

A. K. Cline, ''Scalar- and Planar-Valued Curve Fitting Using Splines Under
Tension, '' Communications of the ACM, vol. 17, no. 4, April 1974, pp. 218-
220.

M. A. Pedersen and D. F. Gordon, ''Normal-Mode and Ray Theory Applied
to Underwater Acoustic Conditions of Extreme Downward Refraction, "'

Journal of the Acoustical Society of America, vol. 51, no. 1, January
1974, pp. 323-368.

L. Brekhovskikh, Waves in Layered Media, Academic Press, New York,
1960.

D. Ludwig, ''Uniform Asymptotic Expansions at a Caustic, ' Communica-
tions on Pure and Applied Mathematics, vol. 19, no. 1, 1966, pp. 215-250.

B. Van der Pol and H. Bremmer, ''The Diffraction of Electromagnetic
Waves from an Electrical Point Source Round a Finitely Conducting Sphere,
with Application to Radio Telegraphy and the Theory of the Rainbow,"
Philosophical Magazine, vol. 24, pt. 1, July 1937, pp. 141-176, and pt. 2,
November 1937, pp. 825-864.

G. A. Leibiger and D. Lee, 'Application of Normal Mode Theory to Con-
vergence Zone Propagation," Vitro Laboratory Research Memorandum
VL-8512-12-0, Vitro Laboratories, West Orange, New Jersey, 30 Novem-
ber 1968.

E. L. Murphy, "Modified Ray Theory for the Two-Turning- Point Problem,"

Journal of the Acoustical Society of America, vol. 47, no. 3, March 1970,
pp. 899-908.

F. R. DiNapoli, Fast Field Program for Multilayered Media, NUSC Tech-
nical Report 4103, 26 August 1971.

Acoustic Environmental Support Detachment, ''Fast Asymptotic Coherent
Transmission (FACT) Model," Office of Naval Research, 1 April 1973.

P22

13.

14.

15.

16.

G. A. Leibiger and D. Lee (This program has not yet been documented.
See reference 9 for an earlier version. )

H. Weinberg, Navy Interim Surface Ship Model (NISSM) II, NUSC Technical

Report 4527, 14 November 1973.
H. Weinberg and R. Burridge, "Horizontal Ray Theory for Ocean Acous-

tics, '' Journal of the Acoustical Society of America, vol. 55, no. 1, Jan-

uary 1974, pp. 63-79.

A. D. Pierce, ''Extension of the Method of Normal Modes to Sound Propa-
gation in an Almost-Stratified Medium," Journal of the Acoustical Society
of America, vol. 37, no. 1, January 1965, pp. 19-27.

NORMAL MODES IN OCEAN ACOUSTICS

D. C. Stickler

Applied Research Laboratory
Pennsylvania State University

The utility of using normal-mode theory to explain acoustic
phenomena when dealing with underwater acoustic problems has
been established. Pekeris used it to predict the results

of shallow-water transmission of explosive charges. This
report discusses applications of normal-mode expansions and
the role of the discrete and continuous spectrum, it provides
a physical interpretation, describes the effect of both proper
and improper or leaky modes, describes the differences arising
from the branch-cut choices, and considers the effect of shear
waves on the pressure field.

Several working computer programs based on normal-mode theory
are compared both by a general description of their capa-
bilities and by their specific treatment of the discrete and
continuous spectral contributions.

BACKGROUND

During World War II C. L. Pekeris became the first to apply
normal-mode theory to problems in underwater acoustics. Since that
time this technique has been employed to explain various acoustic
phenomena. The elementary model used by Pekeris will be used here

to describe some of the properties of normal-mode expansions.

To apply normal-mode theory in underwater acoustics it is nec-
essary to assume that the acoustic parameters depend on the depth
coordinate alone. This means, in particular, that both the longi-
tudinal and shear speeds and the density depend only on depth and
that the ocean-surface and ocean-bottom interface are flat and
orthogonal to the depth coordinate. Specifically, this excludes

sound-speed profiles that depend upon either range or azimuth

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

and environments involving sloping bottoms. Perturbation techniques
can be used to extend normal-mode theory to less restrictive environ-

ments.

Fortunately, in many areas this is an adequate model, and normal-
mode theory has proved successful in explaining various acoustic
phenomena. In those environments where this is not true, other
methods must be employed. All these alternative methods involve
approximations, some of which cannot be listed directly. The validity
of some of these approximations can be tested by comparison with the
"exact" normal-mode representation. These comparisons are certainly
necessary, and they usually yield considerable insight into the nature

of the approximations as well as suggest methods for improving them.

This paper has two points of focus: (1) the physical interpreta-
tion of the concept and techniques of normal-mode expansions, and
(2) the description of those features of the expansion that are the
result of the assumption that the depth coordinate is semi-infinite.
Expanding slightly on this second point, consider the case of an
acoustic wave guide of finite cross section with perfectly reflecting
walls. The normal-mode expansion of the pressure field for this case
consists of an infinite, discrete sum of normal modes. If one of
these wave-guide walls is moved to infinity, then the normal-mode
expansion must be modified, depending upon the behavior of the sound
speed at infinity. Physically, this modification accounts for the
energy that can now be propagated to infinity in this new direction.
The principal effect on the normal-mode expansion is that, in general,
the representation consists of a sum of trapped or proper modes plus

an integral superposition of modes.

This feature depends on the nature of sound speed as the depth

coordinate approaches infinity. If the sound speed is constant

126

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

as the depth coordinate becomes infinite, then the representation
consists of a finite sum of "trapped" modes (there may be none) plus
an integral superposition which can sometimes be approximated by a

sum of "leaky" modes. If the sound speed approaches zero sufficiently
rapidly as the depth coordinate approaches infinity, there are no
trapped modes, only an integral superposition that, as above, can be
approximated in some situations as a sum of leaky modes. Finally,

if the square of the index of refraction approaches minus infinity

as the depth coordinate approaches infinity, then no energy can be
propagated to infinity, and the representation is an infinite sum

of trapped modes.

Note that in the first two examples acoustic energy can be
propagated to infinity. This is reflected in the fact that the con-
tinuous superposition of modes is present, while in the last it is

not.

This paper attempts to develop a more precise meaning and to
provide a physical interpretation for these terms. The basic point
is that the nature of the representation depends on how the sound-
speed profile is terminated. Furthermore, it should be pointed out
that, while in some circumstances one termination is to be preferred

over another, in general, each can be valid and useful.

This paper consists of two parts: a general description of
normal-mode expansions, and a brief summary of some of the programs

in underwater acoustics.

127

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

NORMAL-MODE THEORY

Integral Representation

A typical sound-speed profile in the ocean, shown in Figure l,
has the following special features: it shows the presence of shear
in the bottom, and both the shear and sound-speed profiles in the
bottom are terminated in isovelocity, constant-density half spaces.
Some consequences of this choice for the termination of the sound

profile will be discussed later.

The Hankel transform, chosen for the initial representation of
the pressure field, is useful for two reasons: (1) This approach was
used by Pekeris (1948), Ewing, Jardetsky and Press (1957), and
Brekhovskikh (1960) and, hence, should be familiar to most workers
in underwater acoustics. The alternative representation, based on
Titchmarsch (1946) and described by Labianca in his paper on surface
ducts (1973) is another possibility and, indeed, many of the subtle
analytical properties are best discussed by that method; and (2)
several points about proper or trapped modes, improper or leaky modes
and branch cut integrals, and the physical interpretation of these

terms, seem to fit best in the context of the Hankel transform.

The Hankel representation for the pressure field p at an observa-
tion point (r, z) due to a point harmonic source at (0, Zz) is given

p (r,Z,2) = P (2,2 +k) J, (kr) kdk, (1)

=O)

where ae is the Beroen order Bessel function of the first kind, and

P (2,2 7k) is the transform of p (r,2,Z,) with respect to r. This

128

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER:

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129

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

representation is a cylindrical superposition of waves, and the inte-

gration variable corresponds to the radial wave number.

The integration contour can be taken along the real k-axis if
loss is introduced; if not, it can be taken just below the real axis.
The transform P(z,zZ0 1k) is determined by the sound-speed and density
profiles, pressure-release condition at the ocean surface, continuity
conditions at discontinuities in the acoustic properties, anda
radiation condition. The determination of this function and the
evaluation of this integral constitute the central practical problem
in a normal-mode expansion. In a liquid region (with no shear)
P(z,Z_,k) satisfies
d al dap

0 (z) mG p@) ae

where p(z) is the density and k(z) = w/c(z). The presence of the

2m factor is due to the cylindrical symmetry.

A physical interpretation of the integration contour can be
made in terms of incidence angles as shown in Figure 2. The polar
transformation shown makes it possible to describe the pressure
field as an integration over real and complex incidence angles or,
alternatively, in terms of homogeneous and nonhomogeneous "plane"
waves. The integration over (0, kj) in wavenumber space then
corresponds to "real" incidence angles and the integration over

(ky, co) to nonreal incidence angles.

The discussion of normalemode expansions is simplified by
transforming Equation 1 so that the integration contour is infinite
and the standing-wave component J (kr) is replaced by an outward-
going wave component ie (Gear the Hankel function. The repre-
sentation is given in Equation 3 and the integration contour is

shown in Figure 3. This technique is used by Brekhovskikh (1960).

130

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER:

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SATONY »SINATIINI. Tu

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131

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER

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ANVId *

°¢ omznbta

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1132

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

i (5)
P(xr,Z,2) te P(z,Z 1k) Hy (kr) kdk. (3)

Cc
fe)

. ; ry eae =/
The integral contains two factors KS = ke and Kk” = 7 that

introduce the branch point singularities in P(z,Z > /k). Their
presence is due to the isovelocity termination of the longitudinal

and shear sound speeds.

Integral Evaluation

The evaluation of the integral in Equation 3 can be performed in
several ways. To obtain a normal-mode expansion, however, it is
necessary to identify the singularities of P(Z,Z +k) as a function
of k and to close the C integration contour around these singularities.
For the class of sound-speed profiles described in Figure 1, the
singularities of P(z,Z0,k) are of two types, an infinity of poles

plus two pairs of branch points.

In these ocean models the depth coordinate extends to infinity;
therefore, the representation of the pressure field is always a finite
sum of proper modes (which are defined below) plus a contribution of
the continuous modes. For the profile described in Figure 1, the
continuous modes are represented by an integration around the branch
cuts mentioned above. For other terminations, this contribution can
take a different form. This point will be discussed more completely
below, but, roughly speaking, the continuous modes represent energy
that does not remain ducted, and they will form part of the repre-
sentation of the field when the sound-speed profile allows energy to
be lost to infinity in the z direction. It is not a loss mechanism
in the sense that acoustic energy is transformed to thermal energy

but, rather, it represents acoustic energy radiated to infinity.

133

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

The residue contributions from the poles give rise to two types
of modes: proper and improper. These will be described more
completely below, after the role of the branch points has been

discussed.

Physically, the outward going wave condition in the high speed
sound layer gives rise to one pair of branch points. The other pair
is due to the outward-going wave condition in the isovelocity shear
layer. Each pair of branch points gives rise to a single branch-line-
integral contribution to the field. Furthermore, the choice of the
branch cut influences which modes or pole singularities contribute

to the acoustic field. This point is now discussed.

To describe the differences arising from the choice of branch
cut, it is convenient to examine the Pekeris model and to determine
the differences in the representations that result from the two most
common choices for the branch lines. The statements that will be
made about this model apply, with little change, to the more general

profile.

The Pekeris model, shown in Figure 4, consists of an isovelocity
layer over a high-speed isovelocity half space. There is no shear in
this model. The two common choices for the branch cuts are shown in
the two lower figures. The EJP branch on the left is the branch cut
chosen by Brekhovskikh; on the right is the branch cut chosen by
Pekeris. First, I will discuss the representation arising from the
EJP (Ewing, Jardetsky, Press) branch. The EJP branch is chosen such
that on this sheet Im V ing = ee 20. The negative root occurs on
the second sheet. This means the residue contribution from any pole
on this sheet will eventually decay exponentially with depth and will
represent a mode with finite energy. For this reason, these residue

contributions are called proper modes.

134

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER

INV 1d-*

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STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

An analysis of the integrand for this model indicates only a
finite number of such poles and that they lie between k and k- This
result can be anticipated physically and is not restricted to this
elementary model. Since no loss mechanisms are present, we expect
these proper wavenumbers to be real, to give rise to a standing wave
field in the depth coordinate in the ocean layer, and eventually to
decay with depth. This leads directly to the condition ke < ko <kes

il
Note that these poles lie in the region of real incidence angles.

These proper roots have, in addition, the following properties
and physical interpretations (see Figure 5): the phase velocity ce
satisfies at < ce < Chi that is, the phase velocity in the radial
direction is faster than that in the ocean layer and slower than that

in the bottom.

These modes can also be thought of as being formed by a pair of
plane waves traveling in the plus and minus z directions at an angle
ae with respect to the z axis. This angle fa satisfies, through the
simple polar transformation described earlier,

peal,
> =
Y (C) sin c,/cy,

Recall, further, that for such an incidence angle, the plane-wave
reflection coefficient has a modulus equal to one. That is, at
these angles, no energy is transmitted into the bottom — all of the
energy is trapped in the ocean layer and the fields must, therefore,
decay with depth into the bottom. This is the origin of the term
trapped mode.

Finally, the modes KO near k= (Gizes, ce near c_. or oe near 1/2)

1 ill
correspond to the low-order modes, while those kK = kh correspond to
the high-order modes. The turning points for each of these trapped

modes occur at the interface between the ocean and the bottom.

136

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

Finally, the representation for the EJP branch is given by the
symbolic equation at the bottom of Figure 5: a finite sum of trapped
or proper modes plus an integral around the EJP branch plus an integral
around a semicircular contour at infinity. This latter integral can be
shown for this branch to be arbitrarily small for all observation and

source coordinates.

The representation arising from the Pekeris branch (see Figure 4)
can be thought of as being formed by pushing the EJP branch to a
vertical position. When this is done, some of what had been on the
second sheet of the EJP branch is exposed. That is, in the unshaded
region the condition Im Vik - i < O. Any residue that arises from
a pole in this shaded region will eventually grow exponentially with

depth and, thus, will represent a mode with infinite energy. Such a

mode will be called an improper mode.

An infinity of these improper poles has been found and the
reason will be clear later. Some of the properties of these improper
modes are shown in Figure 6. For these modes, the real part of
ko Ke satisfies kt < kh and their phase velocity in the radial
direction is greater than the phase velocity in the high-speed bottom.
They are sometimes called fast waves. In addition, for the plane-
wave incidence-angle analogy, ee < oe For such angles the reflec-
tion coefficient |R| < 1, and a plane-wave incident at such an angle

will have some of its energy transmitted or leaked into the bottom.

This is the origin of the term leaky mode.

These leaky modes not only eventually grow exponentially with
depth and, hence, do not represent fields with finite energy, they
also have another rather unphysical property: they decay exponen-
tially with range. This decay suggests physically that some absorp-

tion mechanism rather than a radiation-type mechanism is present.

137

NORMAL MODES IN OCEAN ACOUSTICS

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138

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

The pressure field representation for the Pekeris branch is thus
described by the symbolic equation at the bottom of Figure 6. For
this choice the integration at the contour at infinity can be shown
to be zero only if r is sufficiently large or z sufficiently shallow.
This can present serious practical as well as theoretical problems.
In addition, when the contribution of the integral at infinity is not
zero, the sum of the trapped modes is divergent. When this representa-
tion is convergent, it can be used and, furthermore, when it does

converge, we see that

E (leaky) + ve Pekeris = e/ EJP.

BR BR

The foregoing analysis is, of course, not restricted to the
Pekeris model. The simple plane-wave interpretations are model
dependent, but the differences in representations due to the two

choices of branch cut are not. There are two small differences:

@ In the Pekeris model each mode has only one turning
point and it occurs at the ocean-bottom interface.
In a refracting ocean this is not the case. There
may be more than one; however, as in the Pekeris
model none can occur in the isovelocity half-space.

@ The critical-angle concept depends on source loca-
tion and the sound-speed profile, and it is defined
by the grazing ray.

General Comments

In the next several paragraphs I will make several comments
for general profiles, neglecting for the moment the effects of shear.
It is convenient to return to a point discussed in the introduction,
namely, the dependence of the representation on the termination of

the sound-speed profile. If 67) > 0 sufficiently rapidly, then

739

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER:

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u
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HONWad STedMddd

140

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

there are no proper or trapped modes; this has been pointed out by
Labianca (1973) in his study of a surface-duct model. The result
follows directly from a theorem by Titchmarsch (1946). In this case
there is only a continuous superposition of modes. This integral can

be evaluated, when the range is sufficiently large or the z sufficiently
small, by summing the leaky modes — as has been done by Pedersen and

Gordon (1965).

Tee ee >+- 7° as z>, then it is clear that no energy can propa-
gate to infinity in the z direction. For this termination, Titch-
marsch (1946) has shown that there are only trapped modes. Such a

termination has been used by Fitzgerald (private communication).

For the isovelocity termination, numerical examples show that
the branch cut integrals can, in general, be expected to be important
to a range of one water depth and sometimes more. Physically, they
can be expected to be important when there is a constructive inter-
ference of the lateral wave and proper modes. This occurs for a set
of modes near cut-off. An example will be presented later to illus-

trate this point.

When convergence is not a problem, one can ask, "When does a
finite set of the leaky modes offer a good approximation to the EJP
branch?" Numerical experience shows that this sum is not always a
good approximation. This point will also be illustrated in a later

example.

It can be established that the EJP branch decays roughly alge-
braically with range and faster than Wx; thus, it is not surprising
that the sum of leaky modes alone, which decay exponentially with
range, is sometimes a poor approximation to the EJP branch. Returning

to an earlier point, this also suggests why it takes an infinity of

141

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

these improper modes to approximate the algebraic decay associated

with acoustic energy radiated to infinity.

Finally, it is interesting to bring the virtual-mode concept of
Labianca (1973) into this framework. For the profile in Figure l,
the virtual-mode sum is obtained by an approximate integration of the
EJP branch integral. This approximate integration accounts for the
proximity of the leaky poles to the integrand of the branch line

integral.

Effects of Shear

This section is concluded with the description of some of the
effects that the presence of shear introduces into the representation
for the pressure field. These effects are summarized in Figure 7 and,
for completeness, the two corresponding cases neglecting shear are
also included. These are at the top of Figure 7 with the case just
considered being on the right. The case on the top left represents
a case in which the "bottom" speed is less than water speed. It is
not particularly useful since, in any model of the bottom, the sound
speed eventually becomes greater than that in the water. However, for
this case, there are no proper modes and, hence, the representation
consists of either a single EJP-type branch integral or a Pekeris-
type branch plus an infinite sum of improper modes. The convergence
of the improper-mode sum can, again, be guaranteed only when z is

sufficiently small or r is sufficiently large.

Continuing to the cases where shear is present, it is quite
straightforward to show that only when ce is larger than cy Sake
possible to have trapped modes. In the other two cases, one is
either faced with the evaluation of the EJP branches or of the Pekeris

branch and determination of the leaky-wave modes. The convergence of

142

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER:

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(dddVul + HONVYE df4-c

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143

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

the leaky-wave modes still depends on the range and depth coordinate.
When there is refraction in the model, the situation becomes more
complicated. A discussion of the Stonely wave will not be given

here (see Ewing, Jardetsky, and Press, 1957).

A hybrid representation is, of course, possible. For example,
the diagram at the bottom center of Figure 7 might be chosen to model
a sedimentary layer (i.e., the longitudinal sound speed higher than
the water speed but with the shear speed slower than the water speed.)
As remarked above, for this geometry there are no trapped modes; hence,
one representation would consist of two EJP branch integrals. However,
if one chose an EJP branch for the longitudinal speed and a Pekeris
branch for the shear speed, then the representation would consist of
an infinity of leaky shear modes plus a Pekeris-type branch and an
EJP-type branch. Some care must be exercised in this approach be-
cause, while it may be possible to neglect the contribution of the
Pekeris branch, we can expect the EJP branch to yield a contribution

comparable to the sum of leaky shear modes.
BRIEF DESCRIPTION OF EXISTING NORMAL-MODE PROGRAMS

In this section, several working normal-mode programs are

described.

The first group includes programs constructed by Cybulski, by
Kanabis, by Blatstein and Uberall, as reported by Spofford (1973),
and by Newman and Ingenito (1972). These programs all involve a
numerical integration of Equation 2 beginning at the ocean-bottom
interface and using the pressure-release condition at the ocean sur-
face to determine the characteristic wavenumbers KO and the wave

functions P(z,z ,1). The proper modes are summed.

144

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

Bartberger (1973) uses the same numerical integration scheme,
but he determines the proper as well as improper modes. He sums the

proper modes plus a finite number of the improper modes.

Pedersen and Gordon (1965) consider a profile in which re
approaches zero as 1/z and, hence, as mentioned above, is one in which
there are no proper modes. They partition the sound speed in the
upper portion of the sound-speed profile into layers such that the
square of the index of refraction can be approximated by a straight
line and the density by a constant. They determine and sum a finite

number of improper modes.

Kutschale (1970) partitions the sound-speed profile into layers
such that in each layer the sound speed and density can be approxi-
mated by a constant. He allows for shear in any layer. He determines

and sums the proper modes and evaluates the EJP branch integrals.

Beisner (1974) uses a "shooting" technique to determine the

proper modes and wavenumbers, and he sums the proper modes.

Deavenport and Beard (see Spofford, 1973) model the profile as
an Epstein layer. The depth function can then be expressed in terms

of hypergeometric functions. They determine and sum the proper modes.

Leiberger uses WKB techniques to determine the proper modes.

This work is described briefly by Spofford (1973).

Fitzgerald (see Spofford, 1973) partitions the sound-speed
profile in layers in the same manner as Pedersen and Gordon, but he
terminates in a layer in which EGE) >+--°% as z>%, He sums a

finite number of the trapped modes.

Stickler (1975) partitions the sound speed into layers such that

in each layer the sound speed can be approximated by a straight line

145

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

and the density by a constant. He sums the proper modes and evaluates

the EJP branch integral.

Examples

In this section two comparisons are made; they are chosen to
illustrate the importance of the continuous modal contribution.
Consider the profile shown in Figure 8; it is of the type considered
in Figure 1. It is very interesting because, at 50 Hz, there is only
one proper mode, and it is quite near cut-off. In Figure 9, the
transmission loss is shown at 50 Hz for a source at a depth of 20 feet
and a receiver at 40 feet. The lower solid curve represents the
contribution of the single proper mode. Blatstein's calculation (see
Spofford, 1973) for this one proper mode is in good agreement. Bart-
berger (1973) has summed not only the one proper mode but several
of the improper modes. However, for this case, it is seen that the
leaky modes make virtually no contribution. Bartberger's calculation
does not include the corresponding Pekeris-type branch. The upper
solid curve is the sum of the one proper mode plus the EJP branch as
calculated by Stickler (1975). The results of Kutschale (1970), who
sums the proper modes and adds the EJP branch contribution, are seen

to be in close agreement.

This calculation shows two interesting points: 1) The contribu-
tion of the continuous modes can be important to many water depths,
and 2) the sum of the leaky modes is not always a good approximation

to the EJP branch integral.

Figure 10 shows a plot of transmission loss for the same geometry
except now the frequency is 100 Hz. There is still only one proper
mode, the smooth lower curve. The upper solid curve shows the contri-

bution of the proper plus the EJP branch contribution, and the dots

146

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

SOUND SPEED (ft/sec)
4950 5000 5050

SOURCE (20 ft)

RECEIVER (40 ft) es

DEPTH
(ft)
100

BOTTOM
150

200
Figure 8. SOUND-SPEED PROFILE IN SHALLOW WATER

147

NORMAL MODES IN OCEAN ACOUSTICS

STICKLER:

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NORMAL MODES IN OCEAN ACOUSTICS

STICKLER

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149

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

indicate the sum of the proper plus a finite number of improper modes
(neglecting the Pekeris branch). This example shows that sometimes
the improper modes offer an excellent approximation to the EJP branch

integral.

REFERENCES

Bartberger, C., "Normal Mode Solutions and Computer Programs for
Underwater Sound Propagation," Part I - Two-Layered and Three-
Layered Programs, Report No. NADC-72001-AE; Part II - Program for
Arbitrary Velocity Profiles, Report No. NADC-72002-AE, Naval Air
Development Center, 4 April 1973.

Beisner, H. M., “Numerical Calculation of Normal Modes for Underwater
Sound Propagation" IBM J. Res. Develop., pp. 53-58, January 1974.

Brekhovskikh, L. M., Waves in Layered Media, Academic Press,
pp. 454-460, 1960.

Ewing, W. M., W. S. Jardetsky, F. Press, Elastic Waves in Layered
Media, McGraw-Hill, pp. 126-151, 1957.

Kutschale, H. W., "The Integral Solution of the Sound Field ina
Multilayered Liquid-Solid Half Space with Numerical Computations
for Low-Frequency Propagation in the Arctic Ocean," Lamount-
Doherty Geological Observatory, Tech. Report No. 1, 1970.

Labianca, F. M., "Normal Modes, Virtual Modes, and Alternative
Representations in the Theory of Surface Duct Sound Propagation,"
wi. Acoust. Soc. Am. 53, pp LIS 7=1157 7 AGT:

Newman, A. V., F. Ingenito, et al., "A Normal Mode Computer Program
for Calculating Sound Propagation in Shallow Water with Arbitrary
Velocity Profile," Naval Research Laboratory, Washington, NRL
Memorandum 2381, January 1972.

Pedersen, M. A. and D. F. Gordon, "Normal-Mode Theory Applied to

Short Range Propagation in an Underwater Acoustic Surface Duct,"
Je Acoust. Soc. Am. 34, pp. LO5-1187 1965).

150

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

Pekeris, D. L., "Theory of Propagation of Explosive Sound in Shallow
Water," In: "Propagation of sound in the ocean," Geological Soc.
Amer. Memoir 27:1-117, 1948.

Stickler, D. C., "A Normal Mode Program with both Discrete and Branch
Line Contributions," J. Acoust. Soc. Am. 57:856, 1975.

Spofford, C. W., "A Synopsis of the AESD Workshop in Acoustic-Propa-
gation Modeling by Non-Ray-Tracing Techniques," Acoustic Environ-

mental Support Detachment, Tech Note, TN-73-05, November 1973.

Titchmarsch, E. C., Eigenfunction Expansions Associated with Second-
Order Differential Equations, ch. V, Oxford, 1946.

ilsyl

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

DISCUSSION

Dr. F. D. Tappert (New York University): Have you examined the
question of what happens to these branch cuts when you make the para-

bolic approximation?

Dr. Stickler: Not directly, but I think that your restriction
of parabolic method to low angles of incidence roughly corresponds to
reflecting the integration around the branches. Physically that can
be interpreted as integration over the faster phase velocities, which

in turn correspond to the higher modes.

Dr. Tappert: The parabolic equation does have a continuous part
to the spectrum and I wonder where it comes from? From the Helmholtz
equation? It's not a proper mode so it must be either a branch-cut

contribution or a leaky mode.

Dr. Stickler: You mean the spectrum of your parabolic operation

has a continuous mode?

Dr. Tappert: Yes. It may be in the integration along the

semicircular --

Dr. Stickler: I'm not sure there is a one-to-one correspondence.

I don't know.

Dr. R. R. Goodman (Naval Research Laboratory): When doing these
computations one should be aware that the experimentalists can put more
than one hydrophone in the water and one can do some interesting space
and time correlations to look at some of the realities of these various
contributions. I think this is an important point because one can then
begin to design an experiment to look for the types of things you are

talking about.

Dr. F. M. Labianca (Bell Telephone Laboratories): I tend to
agree that there is a continuous spectrum for the parabolic equation,
but let me clarify one thing. Are you referring to the case where there

is no range dependence in sound speed? In other words, where separa-
tion of variables applies?

152

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

Dr. Tappert: Yes.

Dr. Labianca: I agree there is a continuous spectrum in that
case because the depth dependence is going to be exactly the same,
you know, just straight separation of variables on the Helmholtz

equation.

Mr. A. O. Sykes (Office of Naval Research): Would you clarify
Figure 9 for me? There seem to be two groups of normal-mode models

which give different results. Can you comment on that?

Dr. Stickler: Typically, it is, of course, much easier to only
sum the modes that are involved. Carrying out the numerical integra-
tion for the branch-cut integral is a much more expensive proposition
and so usually the branch-cut integrals are neglected or dismissed as

not important at long range. Many times that is the case.

If I had used calculated or summed proper modes, then I would
have made the prediction labelled "ARL discrete." Bartberger summed
the proper and a finite number of the improper modes. They fell on

the other curve.

When I added to the discrete contribution the contribution of
the Ewing, Jardesky, Press type branch, then the transmission is

given by the curve, "ARL discrete plus continuous."

Mr. Sykes: Is the point that some of the improper modes have a
finite contribution which really should be included and so you think

that the upper curve is the better estimate?

Dr. Stickler: Yes, the upper curve is a better estimate.
Figure 9 illustrates several points. First, as I mentioned earlier, a
sum of the improper modes is not always a good approximation to the
Ewing, Jardesky, Press type branch. And it also illustrates that the
Ewing, Jardesky, Press type branch cannot be neglected in some

examples.

153

STICKLER: NORMAL MODES IN OCEAN ACOUSTICS

On the other hand, in Figure 10, which is the same case for
100 Hz, the proper plus a finite sum of the improper modes is an ex-
cellent approximation to the sum of the proper plus the Ewing,

Jardesky, Press type branch.

Dr. Goodman: In this case you have only one proper mode.

Isn't that right?

Dr. Stickler: This case only has one proper mode but it has,

of course, an infinity of improper modes.

154

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION
METHOD IN UNDERWATER ACOUSTICS

Frederick Tappert

Courant Institute of Mathematical Sciences
New York University

A review of the parabolic-equation method in underwater
acoustics is presented. Applications of the parabolic-
equation method discussed here include:

e Short-range (several hundred nm) calculations of
transmission loss

e Calculations of transmission loss in environments
with variable sound-speed profiles and bathymetry

e Calculations of fluctuating acoustic fields in a time-
dependent fluctuating ocean using a model for a random
internal-wave field superimposed on Munk's canonical
profile.

The parabolic-equation method is also used as the start-
ing point to derive theoretical expressions for fluctu-
ations of acoustic fields in random oceans. Using the
mathematical analogy with Schroedinger's wave equation,
two such techniques are described: the first applies the
wave kinetic equation approach to underwater acoustics;
the second applies the Pauli master equation approach to
the same problem.

Theoretical and numerical studies and comparisons to
field data lead one to believe that the parabolic-—wave
equation adequately describes acoustic waves propagating
in real oceans for frequencies between 5 and at least
500 Hz out to ranges of at least 10,000 nm.

BACKGROUND

Leontovich and Fock (1946), two Soviet scientists, were the

first to approximate an elliptic reduced wave equation by a parabolic

155

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

wave equation in the mid-1940s in connection with tropospheric radio-
wave propagation problems. Since then this method has been rather
widely used in radio physics and ionospheric physics (Fock, 1965;

Malyuzhinets, 1959; Barabanenkov, et al., 1971).

My first exposure to the parabolic-equation method was in work
on radar systems involving the simulation of propagation of UHF radar
pulses in a random ionosphere. My former colleague, Ron Hardin, and
I developed computer codes based on the parabolic-equation method to

simulate radar propagation.

When we became involved in underwater acoustics, it was natural
for us to apply these same methods to the subject of low-frequency,
long-range acoustic propagation in oceans. These applications turned
out to be quite fruitful and a number of results have been presented
prior to this workshop (Hardin and Tappert, 1973; Tappert and
Judice, 1972; Tappert, 1974a; Hasegawa and Tappert, 1973, 1974).
Progress has been rapid (Tappert and Hardin, 1973; Tappert, 1974b;
Tappert and Hardin, 1974), and other workers have continued to develop
and apply these methods (Spofford, 1974; McDaniel, 1974; Benthien, et
ai OTA) re

A key factor in the success of the parabolic-equation method is
the numerical technique used to obtain the solution. The parabolic-—
wave equation is solved directly by the finite-difference split-up
Fourier algorithm which makes use of Fast Fourier Transforms to
achieve accuracy, efficiency, and unconditional stability. This
yields a full-wave (all diffraction effects included), fully coupled
(all mode-coupling effects included) solution for the acoustic field
at all depths and ranges. Realistic ocean environments with depth-

and range-dependent sound speed and volume loss, and layered

156

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

variable-depth bottom structure are readily included in the numerical
calculations. Most recently, a randomly fluctuating (in depth, range,
and time) component of the sound speed has been implemented without

difficulty (Flatte and Tappert, 1974).

OUTLINE OF THE PARABOLIC-EQUATION METHOD

The starting point is the reduced elliptic wave equation for the
pressure, p, given by Equation (1) in Figure 1, where r is the
horizontal range, z is the depth, @ the azimuth angle, ko a reference

wavenumber, and n the index of refraction.

Equation (2) relates KS to the reference sound speed, Car and

the angular frequency, w, and n to the variable sound speed, c.

The basic idea behind the parabolic-equation method is expressed
in Equation (3). The pressure is replaced by a slowly varying
envelope function ~ and an outgoing wave represented by the Hankel
function of zero order, Two approximations are then made:
(1) that one 1s in the far field of the source (Equation 4), and

(2) that the angles with respect to horizontal are small (Equation 5).

These lead to a parabolic wave equation for the slowly varying
envelope function , shown in Equation (6). The equation is para-
bolic because only the first derivative with respect to r occurs,

whereas two derivatives with respect to z occur.

By further neglecting the coupling between azimuthal directions
(that is, the derivatives with respect to the azimuthal angle 6),
the two-dimensional parabolic wave Equation (7) is obtained. This is
the basis for all the computer models of low-frequency acoustic

propagation utilizing the parabolic approximation.

57,

TAPPERT:

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

ELLIPTIC WAVE EQUATION:

13 dp ap. dy Go eto :
= — — += +a + =
= ae oe ) + aa2 2 502 + ke ra (Graearts))) Lo (7 z),.0) sp @)
WwW
k = — = const, n = c,/c(r,z,9)
fo) c fe)
fe)
pa (1)
Let D(z onw) = b(r,2,8,0)H. (kK Y)
Approximations: alee kor De Ak (far field)
oy
De ieee << kK |v (small angles)

2
Piggt haadd hlr ge [n2(r,z,0)<1 + ia(r,z,0)] p = 0

Neglect coupling between azimuths:

als

1 920 :

+ — —> + > [n?(x,z,8)-1 + ia(r,z,9)] v= 0

2k 922
oO

Figure 1. PARABOLIC EQUATION METHOD

158

(1)

(2)

(3)

(4)

(6)

(7)

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

From a more fundamental point of view a more rigorous derivation
can be obtained by noting that the basic approximation is analogous
to factoring the elliptic equation into incoming and outgoing waves.
Such a factorization is, in fact, possible, resulting in a pair of
coupled parabolic wave equations, one for the outgoing wave and
one for the backscattered wave. Note that such a formulation could,
in principle, include a description of reverberation. All of the
numerical work to date, however, has been based on the outgoing-wave

parabolic equation.

Since the parabolic equation is not valid near the source, an
asymptotic matching technique is required. Very near the source the
exact acoustic field is known (especially, say, for an isotropic
point source), and this interior solution must be matched to a solu-

tion of the parabolic equation in the far field.

One way to do this is indicated in Figure 2. Manipulation of
Equations (8), (9), and (10) leads to an expression (Equation (11))
for the complex acoustic field at zero range which, when put into the
parabolic wave equation as an initial condition, simulates in the
far field a point source with unit pressure at a range of 1 yard.
Finally, boundary conditions must be specified to solve the parabolic
wave equation. To simulate the pressure release boundary condition
at the surface, an image source, 180 degrees out of phase with the
true source, is introduced (as shown in Figure 2). This forces the

pressure to be identically zero at the surface.

The lower boundary condition is treated by extending the cal-
culation grid beyond the floor of the ocean, as indicated in
Figure 2. In this "mud" region well below the actual seafloor, an

outgoing wave boundary condition is needed. Rather than directly

159

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EAUATION METHOD IN
UNDERWATER ACOUSTICS

SOURCE MODELING — ASYMPTOTIC MATCHING

2
Zee 2
: 2a 2 peg d = | O
Exact: |p| Po ene Sore Gk eee) ~~ me i! : (8)
Parabolic approximation: |p|? = + {y\2 (9)
Apu ee
Wiese re (10)
a )2/w? Sas do
e.g., Wi(zZ,0) = Ave ° = —— . je le)
Vn w i

BOUNDARY CONDITIONS:

IMAGE BOTTOM

IMAGE
SOURCE

SURFACE: W=0

SOURCE
SEAMOUNT
BOTTOM

Figure 2. INITIAL AND BOUNDARY CONDITIONS

160

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

applying such a boundary condition, its equivalent is simulated by
preventing waves that reach so far below the floor of the ocean from
scattering back into the ocean. This is achieved by introducing a
strong volume attenuation which absorbs all the acoustic waves that
reach this subbottom layer, colloquially called mud. This isa
numerical, artificial absorption introduced solely to remove reflected
waves. There is, of course, a corresponding image mud in the upper

ocean.

While substantial analyses have been performed on the validity
of this approximation, it is still a rather open subject and there
have not yet been developed necessary and sufficient conditions for

its validity.

The best way, of course, is to compare it to field data, and
this has been done in a number of cases by myself, and Spofford (1974)
who also compared it with ray and normal-mode results. Such com-
parisons are not conclusive, however, nor are they a replacement for
precise analytical estimates for the conditions under which the

parabolic approximation is valid.

One such analytical approach is to begin with the geometrical
acoustics approximation to the parabolic wave equation (Figure 3).
The exact ray Equation (12) is shown for a two-dimensional stratified
ocean, where z is the ray depth as a function of range r, and 6 is
the vertical angle of the ray (rather than the azimuthal angle, as
earlier). In the parabolic approximation, the corresponding ray
equation is given in Equation (13) and is the same except for the
factor 1/(n cos aie However, they both have as a first integral

Snell's invariant as expressed in Equation (14).

161

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

RAY TRACING:

Exact: d*z = a A ey edn 12
- i dar2 (n cos@)2 2 dz (12)
Parabolic approximation: atz = 1 an? 163
PP ‘ dar2 2 dz (13)
Both give: neacosGs = const (14)
Rays are same except for range scale.
NORMAL MODES EXPANSION:
ivk? + yu 4x
Exact: Pp =D 4,9, (ze fo) O, (15)
a
# lbs so
: : ant ss A
Parabolic approximation Pp uu ata (2 fe) 2k, (16)
Normal modes are the same.
iG 4
Phase error % 8 Ko yr & 27m when r#X10 nm (17)
fe)

Figure 3. VALIDITY OF PARABOLIC APPROXIMATION
(22D, Stratified Ocean)

162

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

Therefore, the rays which can be obtained from the parabolic
wave equation are the same as the rays which can be obtained from the
elliptic wave equation except for a shift in the range scale. This
constant factor (or value of Snell's invariant) can be absorbed into
the range for any one ray, but not for all rays. The approach used
by AESD is to effectively make this change in range scale for those
rays which tend to dominate the acoustic field. In this way, errors

in the parabolic approximation can actually be reduced.

Because the parabolic equation is a wave equation, in the case
of a stratified ocean the solution can also be expanded in terms of
normal modes, as shown in (15) and (16) for the elliptic and parabolic
solutions, respectively. The parabolic equation also has a continuous
part to its spectrum, and the summation implicitly implies an integra-

tion over the continuous part of the spectrum as well.

Both the normal mode eigenvalues, nee and eigenfunctions, Oe
are the same for the elliptic and parabolic equations. However, the
phase velocities (as reflected in the exponential factors in (15) and
(16)) are different. By expanding the square root in (15) and
retaining only the leading term in Uae the parabolic phase velocity
is obtained. The error, therefore, for a single mode can be estimated
by carrying the expansion to the quadratic term, as is shown in (17).
At 100 Hertz for a typical mode in the sound channel, a phase error
of 27 would be accumulated at a range of 10° nm. Hence, if no change
is made in the range scale, as mentioned earlier, a range error of

about 5 percent accumulates for a ray near 20 degrees.

In Figure 4, the parabolic wave equation is rewritten in terms
of a differential operator, A, and a multiplication operator, B,
leading to (20). Note that both n and a are variable quantities

and A and B do not commute.

163

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

k
CU pga aon fie :
Lgaree 2k nee aS n°“ (x,2,0) <1 + ia(r,z,0)| pp = oO (18)
k
pe 1 92 Es fe) 2 .a|
Let A = ak Dae ; B = 5 Neal alo (19)
fe)

abe IN Lk ol.

et i iAy + iB (20)

a eee ney oe el Adr/z qiBAr QiAAr/2 Were) (21)

aoe a tee
iAAx ees 1 | oo ik? Ar/2k | | F vce

(22)

FEATURES:
as Exponential accuracy in z
PAG Second order accuracy in r
3. Exactly energy conserving (when @ = 0)
4. Unconditionally stable
Bye Computationally efficient

6. Readily implemented.

Figure 4. SPLIT-STEP FOURIER ALGORITHM

164

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

The split-step Fourier algorithm is expressed in (21) as the
solution at a new range, r + Ar, in terms of the solution at range r
operated on by a product of three factors. A is a differential
operator which, when carried in an exponent, is difficult to evaluate
by direct methods. But in Fourier space the operator A is simply a
multiplication and therefore this operator acting on a function of
depth can be quickly and accurately evaluated by first Fourier-
transforming the function of depth, then doing a multiplication by a
precomputed and stored phase function, and finally inverting the

Fourier transform (22).

One can prove by the Trotter product theorem of functional
analysis that in the limit as Ar goes to zero, the iterated version
of this does converge in norm (that is, in the space of discrete
functions or functions defined on a discrete grid) to the solution

of the parabolic-wave equation.

Some of the features of this algorithm are listed in Figure 4.
The advantages of this method (listed in Figure 5) are that, without
any extra effort or computation, it can treat range-dependent
velocity profiles, range- and depth-dependent volume losses, and
variable bathymetry (that is, the depth of the ocean can be an
arbitrary function of range). It is easy to solve numerically

by marching in range.

The disadvantages (also Figure 5) are that for very large
angles with respect to horizontal, which sometimes occur with steep
slopes, there are inaccuracies. (Recently, methods have been de-
veloped to reduce these inaccuracies.) Discontinuities require special
treatment (essentially smoothing), but this can be done in a way that

is consistent with the physics and mathematics of low-frequency

165

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

ADVANTAGES
1. RANGE-DEPENDENT VELOCITY PROFILES INCLUDED
Pes RANGE- AND DEPTH-DEPENDENT LOSSES INCLUDED

305 VARIABLE BATHYMETRY INCLUDED

4. EASY TO SOLVE NUMERICALLY BY MARCHING IN RANGE
DISADVANTAGES
lis STEEP SLOPES (LARGE ANGLES) CAUSE INACCURACIES
Ro DISCONTINUITIES OF VELOCITY, DENSITY, AND VOLUME LOSS NEED TO

BE SMOOTHED

3% AZIMUTHAL COUPLING NEGLECTED.

Figure 5. ADVANTAGES AND DISADVANTAGES OF THE
PARABOLIC-EQUATION METHOD

166

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

acoustic propagation. Also, the neglect of azimuthal coupling can be

remedied at the cost of greater computer running times.

In summary, the technique treats a wave equation in an especially
useful way by applying effective and very rapid computational methods
to generate its solution. The basic program is easy to write because
the algorithm is so simple and stable. For production runs on a daily
basis, a highly optimized version of this program is needed, such as
that developed at AESD by using machine language programming and
sophisticated Fast Fourier Transform techniques. The AESD version

has achieved enormous increases in speed over earlier versions.

APPLICATIONS

The following examples indicate the application of the parabolic-
equation method to several problems in underwater acoustics. These
are displayed in terms of iso-loss contours in range and depth from
the effective "source" which may actually be the receiver. The top
figure of each pair is the basic field contoured in 5-dB intervals.
The lower figure represents a range-averaged field with only the 80-
and 90-dB contours shown as light and heavy, respectively. The shaded
regions are either less than 80-dB loss if inside the 80-dB contour,

or greater than 90-dB loss when bordered by the 90-dB contour.

The first example corresponds to a simple pressure-gradient,
or linear, profile in water 16,000 feet deep over a high-loss bottom.
The effective "source" depth is 8,000 feet. Figure 6 illustrates the
field contours for a frequency of 50 Hz. Point C is the location of
a cusped caustic, the two smooth branches of which are migrating
toward the surface and bottom with increasing range. In ray-tracing

programs, the fields in these regions must be found by using uniform

167

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT:

ZH 0S dO AONENOANA ‘LA 0008 LY aounos
WOLLO@ SSOI-HDIH ‘AIIdOuUd LNEIGWYD-gaaNssadd - SUYNOLNOD SSOT-OSI °9 2eAnbTg

my
Lie 1
a
\
ss
x
es
ap 06-

SINOWUOD gD ¢

Depth 0 to 16,000 ft

Tuu QOS O03 0 Sbuey

168

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

asymptotic expansions involving Pearcey functions for the cusp, and
Airy functions for the smooth caustics. In the parabolic-equation

method, such features are treated routinely.

Figures 7 and 8 correspond to the same geometry for frequencies
of 100 and 200 Hertz, respectively. As the frequency increases, the
diffraction effects are reduced and the caustics are more pronounced.
At 200 Hertz, the second cusp, at twice the distance of the first,
is clearly present. It must be re-emphasized that these contours
are not rays. The ray-like patterns correspond to interference be-
tween up- and down-going rays near the surface, and between pairs

of rays associated with smooth caustics.

The preceding three figures correspond to a bottom with high
volume attenuation so that essentially no energy is returned when
it enters the bottom. Figure 9 is for the same case as Figure 8
(200 Hertz) but with a low-loss bottom (simulated by a strong posi-
tive sound-speed gradient and no volume attenuation). Here the
bottom-reflected paths are spectrally reflected and interfere with

the RSR paths distorting the field contours even around the cusps.

The second example corresponds to the slightly more complicated
environment of a bilinear profile. The ray trajectories for a source
in the thermocline segment of the profile are shown in Figure 10,
compliments of Richard Holford of Bell Labs. Note the formation
of smooth and cusped caustics, RR caustics which effectively surface
reflect, and the intersections of caustics. The correct ray treat-

ments for these cases are extremely complex.

Figures 11 through 14 illustrate the field contours generated
by the parabolic-equation method (again using a high-loss bottom)
for frequencies of 50, 100, 200, and 400 Hertz. At the lower

169

TAPPERT:

SELECTED APPLICATIONS OF THE PARABOLIC~EQUATION METHOD
UNDERWATER ACOUSTICS

170

ISO-LOSS CONTOURS - PRESSURE-GRADIENT PROFILE,

HIGH-LOSS BOTTOM

Figure 7.

FREQUENCY OF 100 HZ

SOURCE AT 8000 FT,

TAPPERT:

SELECTED APPLICATIONS
UNDERWATER ACOUSTICS

OF THE PARABOLIC-EQUATION METHOD IN

ISO-LOSS CONTOURS - PRESSURE-GRADIENT PROFILE, HIGH-LOSS BOTTOM

SOURCE AT 8000 FT,

Figure 8.

FREQUENCY OF 200 HZ

TAPPERT:

SELECTED APPLICATIONS
UNDERWATER ACOUSTICS

OF THE PARABOLIC-EQUATION METHOD IN

yes

badge ey!
ely : :
Ra ne
wp git 9 :
fa . .

ne SON Tene

ISO-LOSS CONTOURS - PRESSURE-GRADIENT PROFILE, LOW-LOSS BOTTOM
FREQUENCY OF 200 HZ

SOURCE AT 8000 FT,

Figure 9.

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT

Depth

ATI4OUud YVANITIA wo

SAVY Usa

CNV wa

“OT eanbty

173

SELECTED APPLICATIONS OF THE PARABOLID-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT:

ZH 0S dO AONENOGYA ‘WOLLOd
SSOT-HDIH ‘AIIMOUd UVANITIA YOA SUNOLNOD SSOT-OSI “TT eanbta

SNS REIN UE SASSI SS ASRS AS Ee
wy eS
~
‘
\

AW Y

Teo

174

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT:

ZH OOT JO AONENOGA

WOLLOd

SSOT-HDIH ‘ATIAOUd UYVANITIG YOA SUNOLNOD SSOT-OST

“ZT eanbty

175

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT:

SSOTI-HOIH

I SS
ee Bt
7 { | Ae EP os ~ Lif y,
f gory = tN
AF = re .
: A é NY y.
at j / \.

ZH 00@ dO AONENOGYA ‘WOLLOd
‘“d7T1dOUd UVANITIIG YOA SUNOLNOD SSOT-OST

i; \

"e€T eanbtg

—

176

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

TSO-LOSS CONTOURS FOR BILINEAR PROFILE, HIGH-LOSS

BOTTOM,

—

Figure 14.

alia)

FREQUENCY OF 400 HZ

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

frequencies, the cusped and smooth caustics near the surface are com-
plicated by surface image interference of the diffraction field. Note
the high degree of correspondence between the field contours at 200
Hertz and the illuminated regions according to the ray trajectories

of Figure 10.

The third example is for the so-called canonical sound-speed
profile of Walter Munk (1974). In this case (Figure 15), the source
is on the axis and a high-loss bottom is placed at the reciprocal
depth of the surface to eliminate RSR paths. The two focal regions

on the axis reflect the basic asymmetry of the profile.

The fourth example illustrates effects associated with a range-
dependent sound-speed profile. The entire field is shown in Figures
16 and 17 for the first and second 80-mile segments, respectively.
The profile at the source (again on the axis) persists for the first
60 miles, at which point the axis is rapidly moved up, resulting in
a concentration of energy near the surface. At a range of 120 miles,
the profile rapidly changes back to the original profile, shifting
the surface-concentrated energy deeper and leading to a continuous
shadow-zone near the surface. Invoking acoustic reciprocity, for a
shallow source moving away from an axis-depth receiver, the inter-
mittent convergence-zone behavior of the signal would change to nearly
continuous reception from 60 to 120 miles and then essentially no
reception beyond. This behavior is a direct result of the strong
horizontal gradients which an adiabatic normal-mode approach could

not treat.

The following examples illustrate effects associated with range-
variable bathymetry. Figure 18 displays the field contours for a

high-reflectivity shoaling bottom where initially refracted energy

178

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT

@OUNOS HLdad-SIxXvV WOLLOd SSOI-
‘aTI1dOUd IVOINONVD S,MNAW YOd SAHNOLNOD SSOT

HDIH
-OSI

“ST eanbTa

179

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

HIGH-LOSS

ISO-LOSS CONTOURS FOR RANGE-VARYING PROFILES,

180

FIRST 80 MILES

BOTTOM, SHALLOW SOURCE,

Figure 16.

TAPPERT:

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

181

ISO-LOSS CONTOURS FOR RANGE-VARYING PROFILES, HIGH-LOSS

Figure 17.

SECOND 80 MILES

SHALLOW SOURCE,

BOTTOM,

TAPPERT:

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

PSs IRE SS SSRI ra oc 5 YW SR Senepmeceseeeri ner aapReeO NR TNT

¢ ek % Resse MEN ecast cone
1 Ra Ss NS
Se x SRN
a Seance oe :
a

182

ISO-LOSS CONTOURS FOR LOW-LOSS SHOALING BOTTOM

Figure 18.

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

is converted to SRBR and RBR leading to the so-called megaphone
effect. Figures 19 and 20 illustrate the impact of a high-loss sea-
mount on energy from an axial source and a near-surface source,
respectively. In the first case, the source couples well to the
near-axial modes which suffer little attenuation in passing the sea-
mount. Hence, beyond the seamount, the high-angle modes are stripped
away leaving the very distinct focal regions. For the shallow source
which does not couple well with the axial modes, the seamount strips
nearly all of the energy away. Figure 21 is for the shallow source
where now the bottom is highly reflecting. Paths which before were
annihilated by the seamount now steepen to SRBR going up the sea-

mount and convert back to RSR and RR on the downslope.

RANDOM OCEANS

The final example of the use of the parabolic-equation method
addresses the random ocean problem (Garrett and Munk, 1972; Munk,
1974). The main advantage of the parabolic-equation method is that
it can take into account rapid range variations in the ocean environ-
ment. We now know that there are important random components in the
acoustic sound speed due to internal-wave fluctuations and micro-

fluctuations in the ocean temperature structure.

The following work was begun this summer with Stan Flatté and
Walter Munk. This discussion is merely an introduction to the work
which is covered in detail in subsequent papers (reproduced in these

Proceedings).

The technique is summarized in Figure 22. By adding a time
dependence to the sound speed (23), it can be expressed as a mean
function of depth and range, and a fluctuating function of depth,
range, and time. The refractive index (24) is then a sum of a

deterministic part and a random part.

183

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

TAPPERT

LNNOWWHS HLIM WOLLOd SSOT

|
i

-HOIH dod

HONNOS HLdddad-SiIxw
SHNOLNOD SSO'T-OSI

°6LT eanbta

184

TAPPERT:

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

ISO-LOSS CONTOURS FOR HIGH-LOSS BOTTOM WITH

SHALLOW SOURCE

Figure 20.

185

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD
UNDERWATER ACOUSTICS

oom pea

186

ISO-LOSS CONTOURS FOR LOW-LOSS BOTTOM WITH
SHALLOW SOURCE

Figure 21.

SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

TAPPERT:

c(z,r,t) = c(z,r) + 6éc(z,r,t), (23)

where 6c is a random function of z,r,t

co? c2
2 fe) eo fe) OC UZ tats)
— COC UO - 2——_—
Thee (Zt7g te) eS Gia). c (24)
c
Use quasi-static approximation (© << (Np =)
6 o Ly
k c2
aw AL 92y fe) ( fe) OC\(Zie,
+ + + —\|\s-1 - tt = 2
75x 2k | dz 2 ce a Co ee A

The solution gives

il (ke = (Wet)
jo) (Hey 1) = Wiaizirte) SaaS fe) oO
eet

(26)

° %
€ P(x, 12, /t,) P*(r5,25,t,) ?

Figure 22. RANDOM OCEAN PROCEDURE

187

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

Using the quasi-static approximation in which the frequencies
of the fluctuations are small compared to both the carrier frequency
and the transit time of the acoustic waves over a horizontal correla-
tion length Lo the parabolic wave equation for a random ocean with
time-dependent fluctuations varying in both range and depth is ex-

pressed in (25).

The solution of this equation gives the pressure as a function
of range and depth and time (26). It is a function of three vari-
ables, represented in the form of a complex envelope and a carrier
wave; wW is simply the complex-demodulated envelope which would be
measured experimentally. Hence, w is a quantity that can be compared
directly to experimental measurements of acoustic fluctuations in
the ocean. Typical quantities of interest are correlations of the
pressure at different ranges, depths, and times. This approach has
been carried out numerically, and the results of that calculation
are presented in subsequent workshop papers dealing with both theory and

comparisons with experimental results.

Two additional theories are being developed in connection with
this problem of wave propagation in random oceans. Using the
analogy with the Schroedinger equation, following Pauli, a Pauli
master equation can be derived using normal modes (Figure 23)
(Agarwal, 1973). The envelope y is represented as a sum of normal
modes (28) with random coefficients ane A density matrix (29) is
defined as the correlation between normal-mode amplitudes, and
coupling coefficients (30) between normal modes are used to represent

the effects of the randomly fluctuating component of the sound speed.

Transition probabilities (31) are developed and finally a master

equation (32) involving only the diagonal elements of the density

188

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

No. ‘al modes:

1 a, be ( cS
a, 922 * 2 \ezzy ~ 1) Yn 7 *nYn ei
wizr) = D> alr) vi (z) em ai (28)
n

Density matrix:

= *
Pam (¥) <a, (x) ax (x)> (29)
Coupling coefficients:
KS
= — *
Vm z fox v* (2) 6c(z,r) v (2) (30)
fe)
Transition probability:
_ e _ 2
a an <|¥ Ok, k | » (31)
Master equation:
2 - &
ox Pnn ~ m Yam (Pram oa fee)

Figure 23. PAULI MASTER EQUATION PROCEDURE

189

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

matrix is obtained. These elements are simply the squared amplitudes

of the normal modes.

Everything in Equation (32) is known and can be numerically
obtained using computers. Comparisons are then possible between
these normal-mode results and either numerical experiments (using

the parabolic equation) or field data.

Another approach, outlined in Figure 24, 1s based on the wave-
kinetic equation, basically applying transport theory to acoustic
propagation of random oceans. The Wigner phase-space distribution
function f defined in (33) is introduced where f is quadratic in the
complex demodulated signal y and hence depends on depth z and range
r as well as vertical angle 6. The ensemble average F (36) satis-
fies the integro-differential equation (37). This equation, which
describes the evolution of the ensemble average Wigner distribution
function (Tappert and Besieris, 1971; Besieris and Tappert, 1973),

is essentially the covariance of the pressure.

Again, everything in this equation is known in terms of the
fluctuations. It has the form of a classical radiation transport
equation and numerical techniques may be used to solve it. The
virtue of this approach is that it leads directly to the ensemble-
average acoustic field (and hence mean intensities) not just at one

point but at two points.

Figure 25 shows a simple example of this method, applying a
diffusion approximation. The correlation function of pressure at
two depths is obtained (41) as an exponential, and (42) gives the
coherence length in depth as a function of range. This is a definite
prediction of the theory that can be compared to either numerical

experiments (for example, using the parabolic equation) or field data.

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

Wigner distribution:

k
£(z,0,r) = = foe eros v(z 15 > 2x} v(e = > 2 | (33)

where
freon ao = |(z,r)|? (34)
freon dz = |W(0,r) |? (35)
F(z,0,r) = Cf£(z,8,r)> (36)
oF . oF , 1 es oF
P) fe)
— + —— ae eee — a ——
or a gz Es 1) 36
= fevrwie,9,0" [F(z,0',r) - F(z,®,r) ] C37)
where
a ae ee) ce
2 pe ee SOS
dr a dr 292 (2s ()
and
= La? Bonne
w(z,6,0")-= 2m k Tfz,6 = 6", — 6° - > 6" (39)
fe) 2 2

Figure 24. WAVE KINETIC EQUATION

ALie yal

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN

UNDERWATER ACOUSTICS

(o
oF oF iy 4g) fe) oF r) oF
Sees a ae oe eat
ar + © 92 2 dz (3 1) Y 56 Pe) a6
Take ¢ = const i D = const
1 it -£ x2 (92> 22
Cw*(z + 5 21+) W(z - 2 Zar): (=). 672 6
L

aL Vv
Z coh v my Kez ~ k,(8e/e) vigr

Figure 25. DIFFUSION APPROXIMATION

192

(40)

(41)

(42)

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

REFERENCES

Agarwal, G. S., "Master Equation Methods in Quantum Optics," in
Progress in Optics, Vol. XI:1-76, North-Holland, 1973.

Barabanenkov, Yu. N., Yu. A. Kravtsov, S. N. Rytov, and V. J.
Matanskii, SOV hus. Uspekhi, IS 55 (1971)

Benthien, G. W., D. F. Gordon, and L. E. McCleary, J. Acous. Soc.
Amer., 55:S45 (1974); also private communications with D. Gordon
and others at NUC.

Besieris, I. M., and F. D. Tappert, J. Math. Phys., 14:1829 (1973).

Flatté, S., and F. D. Tappert, JASON Report, Stanford Research Inst.,
1974.

Fock, V. A., Electromagnetic Diffraction and Propagation Problems,
Pergamon Press, N. Y., 1965.

Garrett, C., and W. Munk, Geophys. Fl. Dyn., 2:225 (1972).

Hardin, R. H., and F. D. Tappert, ‘SIAM Rev. (Chronicles), 15:423
(973):

Hasegawa, A., and F. D. Tappert, Appl. Phys. Lett., 23:142 (1973);
2371) (974).

Lecnt.vich, M., and V. Fock, Zh. Eksp. Teor. Fiz., 16:557 (1946).
Malyuzhinets, G. D., Sov. Phys. Uspekhi, 69:749 (1959).

McDaniel, S. T., J. Acous. Soc. Amer., 55:S45 (1974).

Munk, W. H., J. Acous. Soc. Amer., 55:220 (1974).

, JASON Report, Stanford Research Inst., 1974.

Munk, W., and S. Flatté, Proceedings of this Meeting.

Spofford, C. W., J. Acous. Soc. Amer., 55:S34 (1974); also
private communications with R. Buchal and H. Brock at AESD.

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

Tappert, F. D., in Lectures in Applied Mathematics, Vol. 15, (ed.)
A. C. Newell, Amer. Math. Soc., Providence, R. I., 215-216, 1974.

;, Jie Acous. Soc. Amera,( 557S34)(1974)..

Tappert, F. D., and I. M. Besieris, Proc. Inter. Symp. on Electro-
magnetic Wave Theory, Tbilisi, USSR, Sept. 1971.

Tappert, F. D., and R. H. Hardin, in AESD Tech. Note 73-05, (ed.)
Ge Wa Spofttord, (ONR, Avilington,! Vae, Now. 1973.

; , Proc. Eighth Inter. Congress on Acoustics,
London 1974, p. 452.

Tappert, EH. Ds, and ‘©. Ni. Judice,, Physi. Rev. Lett., 29): 1308 W972)

DISCUSSION

Dr. J. B. Hersey (Office of Naval Research): First of all, I
think our speaker is to be congratulated on an absolutely brilliant
performance. I have been very excited to see some of the nagging
problems of ocean acoustics, not necessarily finally solved but yield-
ing some very, very intriguing and encouraging results. Congratula-

ELONS),» Sasi
Dr. Tappert: Thank you.

Mr. E. D. Garabed (Naval Air Development Center): You mentioned
that in this parabolic-equation method there was a limitation on
angles that it can be used for. Can you give me some idea as to what

that angular limitation is, in degrees?

Dr. Tappert: Roughly a 5 percent error at 20 degrees is intro-
duced in the ray periods or modal phase velocities. The significance
of this error really depends on what you want to measure, or what you
want to get out of the calculation. Some things are computed more

accurately than others.

194

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

For example, if you only want transmission loss, it doesn't
really matter whether you have an error in the phase. But if you
want to do beamforming with the complex signal, then you need accurate

phase information as well.

There is no universal simple answer to your question. We have to
make more computer runs, compare with more data, and do more analysis
before we fully understand all the limits of the method. It is a
flexible method. It is not just one simple formula that you do once
and for all. There are ways to improve and extend and refine this

parabolic-equation method.

Dr. H. Weinberg (New London Laboratory, Naval Underwater Systems
Center): If I understand correctly, you use a virtual source to take
into account the free surface. Would it be easy to take into account

surface loss or its equivalent?

Dr. Tappert: One would think so, and I have struggled hard to
find a way to do it, but with the algorithm that I described it seems

to be difficult to relax the flat-surface boundary condition.

Dr. Weinberg: I don't see why it is more difficult for you to
treat the free surface than some sort of boundary condition. Is that

because of the algorithm you chose?

Dr. Tappert: It is because of the Fast Fourier Transforms. With
other algorithms it would be easier to introduce surface losses and
surface scattering. And I really do encourage others to look into
other algorithms. There is nothing magic about this one. I am con-
vinced that it is unusually efficient and effective and accurate, but
again it would be worth knowing just how much better it is than other

possible numerical algorithms.

195

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

Dr. S. W. Marshall (Naval Research Laboratory): I am getting
back to times for computation again. Can you tell me whether you
use software or hardware FFTs and what your computation time in long

runs is?

Dr. Tappert: Yes. I use software FFTs, but coded in FORTRAN.
At AESD they have a Compass-coded FFT. The difference is about
50 percent. You can in principle achieve gains of an order of magni-

tude by using hardware FFTs.

On a machine like the UNIVAC 1108 or IBM 370/165, without the
fluctuations in the ocean, it takes roughly the same amount of machine
time to compute the acoustic field as it takes the acoustic field
to advance, which is roughly one mile per second, so if you are going

100 miles, it takes roughly 100 seconds.

Dr. R. M. Fitzgerald (Naval Research Laboratory): Regarding the
approximation in small angle for the parabolic-equation method, I
think you can show that the approximation is one in which the angles
are restricted to a small cone but the direction of the cone is

arbitrary.

Dr. Tappert: That is very true. Yes.

Dr. Fitzgerald: And in that way you can overcome the limitation
now that you cannot treat steep rays. You do it by using separate

cones and linearly superimposing the results.

Dr. Tappert: The problem I had in trying to work that out is
how you superimpose. You certainly can take a cone of angles that
is not oriented horizontally. For example, if you want to do the

bottom bounce experiment, you take a cone going down and then it is

196

TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN
UNDERWATER ACOUSTICS

quite accurate within that cone. But I could never see a really good
way to connect different cones without getting an interference pattern,
a spurious interference pattern, where they connect. But if you have

a way to do that, it would be quite an improvement.
Dr. Fitzgerald: I do.

Dr. Tappert: Good.

197

he
+
‘
ee

CALCULATION OF THE EFFECT OF INTERNAL
WAVES ON OCEANIC SOUND TRANSMISSION

Stanley M. Flatte

Frederick D. Tappert

Reprinted from the Journal of the Acoustical Society of America
Vol. 58, No. 6, December 1975

Calculation of the effect of internal waves on oceanic

sound transmission

Stanley M. Flatté
University of California, Santa Cruz, California 95064

Frederick D. Tappert

Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

(Received 22 August 1975)

The signal received by a hydrophone in the ocean many kilometers from a steady sound source fluctuates
dramatically due to variations of the speed of sound in sea water. By inserting an empirical model of
internal-wave-generated sound-speed variations into an acoustic-transmission computer code, we have
shown that internal waves cause significant variations in sound transmission at 100 Hz, comparable in size
and frequency to the variations observed in field experiments. We have also studied the usefulness of

vertical hydrophone arrays.

Subject Classification: 30.25, 30.82; 28.60.

INTRODUCTION

Acoustic transmission in the ocean is profoundly af-
fected by the dependence of the speed of sound on ocean-
ic depth, range, and time. The speed of sound, in turn,
is determined by oceanographic quantities; pressure,
temperature, and to a lesser extent, salinity.

In a typical 4-km-deep ocean, the sound speed c has
a minimum as a function of depth z at about 1 km, with
a value close to 1500 m/sec. Values of c at the surface
and bottom are a few percent higher than at the mini-
mum, From an oceanographic point of view the mini-
mum is due to the competition between the drop in tem-
perature and the rise in pressure as one descends in
the ocean. The minimum causes sound to be refrac-
tively contained in the volume of the ocean, resulting
in a “sound channel, ” and making possible sound trans-
mission over thousands of kilometers at frequencies
below 1kHz.! (Higher frequencies are absorbed. )

The sound-speed profile c(z) varies with geographic
position. For example, the sound channel minimum
rises as one moves toward colder Northern waters. In
addition, the depth of the ocean changes due to the to-
pography of the bottom of the sea. Hence any acoustic
transmission experiment over hundreds of kilometers
or more will be subject to a range-dependent sound-
speed profile. A great deal of work has gone into map-
ping the expected differences in transmission due to
differences in geographical location.”

The strongest time variation of c(z) occurs as a re-
sult of seasonal changes in temperature. These long-
term time variations have also received considerable
attention, although in principle the changes between
winter and summer are no more difficult to deal with
than a significant change in geographical position.

Any experimenter who has done a long-range acoustic
transmission experiment can attest to the fact that con-
siderable (5-—30-dB) variations in signal are observed
over periods ranging from a few minutes to several
days, with several hours being typical. When surface
interactions are absent or have been filtered out, and

201

for fixed source and receiver, these fluctuations must
be caused by variations in the sound-speed field through
which the sound passes, and the sound-speed variations
must have an oceanographic origin. Yet until now al-
most no quantitative connection has been made between
these “rapid” acoustic variations and any realistic
oceanographic phenomenon. Over the years, however,
there has been much speculation and order-of-magni-
tude correlation with internal-wave motions. *

The ocean contains a random field of internal waves,
with periods ranging from 3; to 24h. The intensity of
these waves has been estimated from measurements of
temperature and current fluctuations in the ocean, and
the associated effect on sound speed has been calcu-
lated.**® The root-mean-square sound-speed fluctua-
tions due to internal waves is at a level of 10%, two
orders of magnitude below the variation which causes
the sound channel. °

In this paper we demonstrate that the internal wave-
field in the ocean causes significant fluctuations in
long-range acoustic signals, comparable in size and
period to those observed in field experiments. Our
method of calculation involves a computer code (devel-
oped by us) which propagates CW acoustic signals
through a sound-speed field that depends on both depth
and range. The code calculates the random part of the
sound-speed field from an internal-wave spectrum,
and simulates the time variation by stepping the inter-
nal wavefield in time, and propagating the acoustic
signal at each time.’ We have used a frequency of 100
Hz.

In this paper we also demonstrate the usefulness of
a vertical array of hydrophones in reducing intensity
fluctuations in long-range acoustic transmission.

Other work relating acoustic fluctuations to internal
waves has been in progress simultaneously with ours.
DeFarrari considered only one internal wave, the in-
ternal tide, rather than a full spectrum.® Porter ef al.
considered a full spectrum of internal waves, but used
a thin-layer model for internal waves as they affect

acoustic signals.®° Their model cannot be complete as
it fails for rays whose turning point occurs within the
thin layer. We consider a full spectrum (excluding
tides) and treat the acoustic—internal-wave interaction
within the full volume of the ocean.

The remainder of the paper is organized as follows:
Section I describes the sound-speed field derived from
the internal-wave spectrum, and its computer realiza-
tion. Section II describes the acoustic propagation
method (which depends on the parabolic equation ap-
proximation) and its computer realization. Section III
presents our quantitative results. Section IVis a sum-
mary and conclusion. The Appendix describes our
method of vertical beamforming.

1. OCEAN SOUND-SPEED STRUCTURE
A. Deterministic profile

On the scale of the depth of the ocean (4 to 5 km) the
sound speed as a function of depth z is determined by
the gross behavior of the density, temperature, and
salinity. We use the profile derived by Munk!” whose
input is an exponentially decreasing density gradient.
The resulting “canonical” profile is

Cop(Z)=¢y{1 +e[e7"- (1- n) |} )

where 7=2(z—z,)/B. Note that ccp(z) has a minimum
at z,, that the width of the minimum is B, and the de-
viation of the sound speed from the minimum value c,
is of the order «.

Figure 1 shows Cop(z) for the typical (though not uni-
versal) values we have chosen for the parameters: z,
=1000 m, B=1000 m, €=0.57x10%, and c,=1500 m/
sec.

DEPTH — km

1.48 1.50 52, 1.54 1.56 1.58
SOUND SPEED — km/s

FIG, 1. Deterministic sound-speed profile as a function of
ocean depth (Canonical Profile). The value of c at the mini-
mum (z4=1000 m) is c(z4)=1500 m/sec.

202

One must point out that realistic ocean profiles in
most cases have significantly different behavior from
this general form. For example, within a few hundred
meters of the surface a mixed layer usually results in
a lowered sound-speed gradient. However, we ignore
these details in our present treatment.

B. Internal waves

The density gradient in the ocean leads to the pos-
sibility of waves traversing the volume of the ocean
just as the density discontinuity at the surface leads to
the possibility of surface waves. The density gradient
is usually presented in the form

be ~\ 1/2
N(z)= SS =)
Po 92 7
where N(z) is called the local stability (Brunt—Vaisala)
frequency.

Following Garrett and Munk, ° we assume a stratified
ocean with

N(z)=N, 7/2,
where N,=3 cycles/h.

Let w(r, f) be the vertical component of fluid velocity
at position r and time ¢. It can be shown" that w sat-
isfies the equation

a
oF (v?w) +N?(z)V2w = On

The eigenmodes of this equation can be found by taking
w= Wi, k, z) eilkyxtkgy-w (i,k) t] F

where k=(k%4+k3)/* is the horizontal wavenumber.
Substituting and modifying our equation to account for
the rotation of the earth, we find

aw N 2(z) = w*

—- +|—-—7—| #? w=

az? +f w= wi Usa

where w; = inertial frequency =(2 cycles/day) sin(lati-
tude). We will use w;=1 cycle/day. Boundary condi-
tions are W(z)=0 at surface and bottom (assumed
flat). 1

A particular mode, characterized by mode number 7
and horizontal wave number k, will have a vertical ve-
locity profile given by W(j, k, z) and a definite frequen-
cy w(j,k). Because every fluid element moves with the
same frequency, the vertical displacement ¢ of a fluid
element from its equilibrium position for a single mode
will also be proportional to W(j, k, z). The sound-speed
fluctuation 5c is related to the displacement ¢ by°

bc =Cy EN*2)E < N2(z)W(j, k, 2).
Several examples of the sound-speed profiles due to

particular modes are shown in Fig. 2.

The sound-speed fluctuations caused by a full internal
wavefield may be represented as a linear superposition
of eigenmodes, leading to

bees oe G(j, Ry, Re) N*(z) WJ, B, 2) ettnrtane
IrR yy Ro

K = 0.5 cycles/km

Realistic internal-wave spectra have significant intensities for
horizontal wavenumber less than about 0.5 cycles/km. Note
that the major internal-wave contributions to sound-speed
fluctuations occur at depths less than 1 km,

where the summation sign means integration over the
continuous variables k, and k,. We normalize W(j, k, Zz)
so that

[™ nteywi, k,2)dz=1,
9

where Z,,, is the depth of the ocean.

The numerical difficulty in projecting this three-di-
mensional field onto the two-dimensional vertical plane
used in the acoustic propagation code has caused us to
consider a simplified version of the internal wavefield
where internal waves propagate only in (or opposite to)
the direction that the sound waves propagate. In addi-
tion we combine real and imaginary parts to reduce
fluctuations in the overall energy in the internal waves

203

as a function of time:
5Cyw = (ReA +Im4A)

and

Re > AG, k)N2(z)W(j, k, 2) eter~e eB
fk

where * is the horizontal range.

The A(j, k) are complex Gaussian random variables.
From a synopsis of diverse oceanographic measure-
ments, Garrett and Munk® have proposed the following
model:

(A(j, k)) =0

(A(j, R)A*( 3’, R')) = 8 y;-5pn- X B7H( J) BC, Rk),
where

H(j)=6/(nj)’,

B(j, k) = (2/1) kjk? /(k? +k3),

ky =(1/B) (wi/No) 3 «

The spectrum is normalized so that
Dif HG)BG, #)dk=1 and { B(G, b)dk=1.
F | ‘= =

From the above equations it can be shown that

(reser

But Garrett and Munk® have shown that

Hence 62 =y"B/3 where y is a measure of the fractional
sound-speed fluctuations due to internal waves. From
Ref. 4 we have y=4.8107.

It is useful to point out a few properties of the dis-
persion relation and the spectrum. The frequency
w(j, k) varies between the inertial frequency (1 cycle/
day) and N, (3 cycles/h). Frequency increases with in-
creasing values of k and decreasing values of j.

For a fixed mode number j, the function B(j, k) gives
the relative contribution from each value of k. The
peak of the & distribution is at k,, The function H(j)
gives the overall contribution from each mode number
j; the gravest mode (j=1) has the largest contribution,
with other modes decreasing as 1/j*. The relative in-
tensities of the various modes are shown in Fig. 3.

Note that the magnitude of the sound speed fluctua-
tion due to internal waves is 6c/c~107', a factor of 100
below the magnitude of the deterministic structure.
Also the spatial behavior of the sound-speed variations
due to internal waves is of the order of a few hundred
meters vertically and several kilometers horizontally.
Figure 4 shows some typical sound-speed profiles due
to internal waves.

C. Final expression for sound-speed structure

c(r, t)=Cop(z) + 5c yy

(I A(j,k) 12) —dB

24
-50 :
0 0.1 0.2 0.3 0.4 0.5
k—cycles/km
FIG, 3. Internal-wave spectrum as a function of mode number

j and horizontal wavenumber k. Although large mode numbers
contribute very little to the overall spectrum, they are crucial
to understanding acoustic effects, since their vertical struc-
ture allows them to act as a scatterer of acoustic energy more
readily than the relatively structureless low modes.

D. Numerical realization of the internal-wave model

M. Milder’s program (ZMODE)” was modifed to
numerically generate the eigenfunctions W(j, k, z) and
frequencies w(j,). Modes with 1=j< 24 were in-
cluded. Values of k ranged from — 0.5 to 0.5 cycle/km
in 254 equal steps. The 6096 different A(j, k) were
generated according to a Rayleigh probability distribu-
tion in amplitude, and variance given by the spectrum
described in Part B. The phase angle of each A(j, k)
is randomly generated in the region 0 to 27. Using the
above equations, our code can then generate the sound
speed at any point in space and time.

Il. PROPAGATION OF ACOUSTIC SIGNALS USING
THE PARABOLIC EQUATION METHOD

A. Introduction

The parabolic equation method was originally devel-
oped by Leontovich and Fok in 1946 to study long-
range propagation of radio waves in the tropospher
This method was introduced into the field of underwater
acoustics by Tappert in 1972 and a computer program
based on this method was developed by Tappert and
Hardin to solve acoustic propagation problems of inter-
est to the Navy. 115

e, 38

204

B. Approximations and ranges of validity
The wave equation for acoutic pressure p(r, t) is

92
Vp - A a =0.

Our knowledge that c varies from a constant only by
very small amounts, and that the variations are slow
compared with the acoustic frequency allows us to use
the following expression in cylindrical coordinates for
the pressure (far from the source):

eilkor-wt)

p(7, H=¥, (7, 2, 0) ;

where the reduced wave function ¥ is labeled by the
time ¢t, because the 5c/c structure of the ocean is dif-
ferent for different times. Substituting in the full equa-

(a)

DEPTH — km

: |
(c)

eo =a
2 =
3 =
4 | |
-2 -1 ie} 1 2

8C x 2500

FIG. 4. Sound-speed profiles induced by a full spectrum of

internal waves at a particular instant of time. Let 7 be the
range from some arbitrary point, then (a) r=0, (b) r=14 km,
and (c) r=28 km.

tion, neglecting time derivatives of V and terms of or-
der 1/(k,r)*, we find
ey 1 ev ath

aw 5c
; 2 =
ore +P Ie tage + 2ikey — ako We=i0

where k,=w/c,, and we have assumed that 6c/c <1,

The key to the parabolic equation method involves the
following additional physical approximations, based on
the structure of 5c/c:

if kjL, > 1, where L, is the vertical scale of sound-
speed variations. This condition is equivalent to re-
taining only relatively forward scattering, which re-
sults in small changes in VW over an acoustic wave-
length. It is valid if the objects off which the acoustic
waves are scattering have sizes which are much larger
than a wavelength; and

ey 1 av aw
<—
Q az?’

which is true because the canonical profile, which has
100 times the sound-speed fluctuation than the internal
waves, affects the z coordinate only. More important-
ly, however, the internal-wave gradients in the verti-
cal are an order of magnitude greater than the horizon-
tal. The approximate wave equation is therefore
3 aw
Sa ae (1)

As a result of our approximations, we have neglected
all azimuthal correlations. Thus we cannot study azi-
muthal fluctuations. We can study fluctuations that
can be observed in a single vertical plane, where azi-
muthal correlations have a small effect.

To summarize the approximations required for this
parabolic equation to be valid we have the following
quantities not yet defined: w,y=largest frequency in-
volved in the internal wave spectrum, ~3 cycles/h;

Ly =minimum horizontal scale of sound-speed fluctua-
tions, ~ 1 km due to internal waves; Ly=minimum ver-
tical scale of sound-speed fluctuations, ~ 200 m due to
internal waves. Validity of the parabolic equation re-
quires: wW>ww; kopy>> 1; Ly>Ly; andkLy>1. All
conditions are well satisfied in our case, where w=100
Hz.

C. Numerical realization

We solve Eq. 1 by the “split-step- Fourier” algorithm
of Tappert and Hardin. !® Given W(r, z) we find the wave
function at a new range from the following:

W(r 4dr, 2) = Fe idar $e? Wr, z)]},

where B=-k,5c/c and A =(1/2k,)8?/az?.. Thus A and B
are operators in z space (A being the Fourier transform
of A) and S is a fast Fourier transform operation. 1”
This algorithm is fast and very stable since the total
acoustic energy J| ¥|*dz is exactly conserved as a
function of range when absorption is absent.

The FFT was used with 512 elements over a 4-km-

205

deep ocean, and the range step has been chosen as 0.5
km,

D. Acoustic source and boundary conditions

The acoustic field may be started with any function
of depth (0, z). A point source at z=1000 m (the depth
of the sound channel) with unit strength at one yard has
been modeled by an asymptotic matching technique
which prescribes the appropriate initial value,

The ocean surface has been treated as a perfect
pressure-release boundary so that ¥(7,0)=0. This is
accomplished through the use of a fast sine transform?®
for the operation 5.

In order to model a completely absorbing ocean bot-
tom, a gradual loss of amplitude is imposed on ¥(z) as
Zz nears the ocean bottom. The functional form of the
imposed loss at each step is the factor

& — Zmax 2
L(z)=exp| — a dr exp- Gras
with a=0.05/m and B=0. 04 Zmax.

This form effectively stops any acoustic energy from
penetrating below about 500 m above the bottom. Even
this attempt at acoustic impedance matching does cause
some reflection off the bottom at an extremely low in-
tensity level.

Il. RESULTS

When an explosion is detonated deep in the ocean, a
series of sharp reports are heard at ranges up to sev-
eral thousand kilometers. The fact that each separate
sound arrives without being dispersed in time implies
that a geometrical-optics view of sound transmission
in the ocean must have a great deal of merit. Ray
tracing is a well-established technique for determining
the character of oceanic sound transmisssion,

Figure 5 shows the ray paths where the sound speed

\ ZL
ath U7

’,

WS
Nest y/
Ne we

DEPTH — km
nN

0 10 20 30 40 50 60 70
RANGE — km
FIG. 5, Ray paths for the canonical sound-speed profile given
in Fig. 1, with a source on the sound-channel axis,

20 km 60 km 100 km 250 km
mon [3 T = | =o1: =m) | T XSL ar]
aes } —
L par, Ul ioeacri ie a 10°
80 4
ail. = fis 1 — 1 = =i 4
a = Sn a L L | FIG. 6. Transmission-loss time se-
0 60 eae aia eS ries at several ranges and depths.
| . 5 Each section shows the intensity that
2 go L iu =| = would have been observed by a single
3 hydrophone taking data once an hour
= for 128 h. Each column represents a
5 a 4 fle Le sail 1 fl aft i particular range from the CW source,
2 — ee [ Each row represents a particular ray
= 60 at ge ail followed from the source, (Positive
8 Os AXIS angles correspond to downward rays
x | RAY from the source.) That is, at each
80 ii la range, the depth is chosen as the depth
that particular ray passes through at
re ' 2 \ of \ 1 ' that range.
; cok
80 b 4
at ca ne m = l L 1
50 100 50 100 50 100 50 100
TIME —h

is given by the canonical profile!° and the source is on
the sound axis (z,=1000 m), It is apparent that after a
few tens of kilometers the sound arriving at various
points has a complicated directional character due to
multiple paths. For example, at 60 km on the axis
sound should arrive from three well-separated direc-
tions. Note also that our absorbing bottom at 3.5-km
depth prevents any surface or bottom reflected energy
from propagating beyond about 20 km.

It is possible that some of our results for acoustic
signals traveling through internal waves may be under-
stood in terms of internal-wave effects on individual
rays. It will be well to remember, however, that be-
yond the 20-km range a single hydrophone will in most
cases receive more than one ray from the source.
This multipath effect is crucial to understanding long-
range fluctuations.

Figure 6 shows the computed transmission loss as a
function of time at several ranges for 100-Hz acoustic
signals traveling through the internal wavefield. The
point source is at a depth of 1000 m. Each row shows
results for a particular ray which has been followed
from the source by integrating Snell’s law (e.g:, the
hydrophone at 100-km range for the 6° ray is at the
depth corresponding to the 6° ray at that range). The
1-h time steps clearly undersample the fluctuations,
but the general character of the series is clear. We
see that internal wave sound-speed fluctuations cause
5-30-dB fluctuations in received intensity, comparable
in size to those observed in field experiments. *

206

We have used a vertical beamformer (see Appendix)
to separate the different ray arrivals at various ranges
and depths. Figure 7 shows a time history of one
beamformer output. A single hydrophone would co-
herently add the many rays, each of which are seen in
Fig. 7 to vary in direction and intensity. If the peak of
the ray of interest is chosen at each time, then a time
series for the intensity of that ray can be plotted.
Figure 8 shows time series for the particular rays
corresonding to the single hydrophone results in Fig.
6. It is evident that the fluctuations of a single ray are
considerably muted compared with those of a single
hydrophone which is subjected to a coherent addition
of all rays.

Figure 9 shows the rms intensity variation as a func-
tion of range for four rays. {[{(10log/)*) — (10 log1)?|!/?
is plotted. } Both single hydrophone and ray-peak re-
sults are plotted. The reduction in fluctuation that re-
sults from selecting a ray peak is clear. In addition,
the smoothness of the rms value as a function of range
for the ray peak gives grounds for hope that a simple
single-ray theory might be used to predict the range
dependence of the fluctuations.

IV. SUMMARY AND CONCLUSIONS

We have developed a parabolic-equation acoustic
propagation code which sends CW sound signals
through a time-dependent random internal wavefield
superimposed on a deterministic sound channel.’ The
output is the complex pressure field as a function of

40 SRLS
no internal
60 waves

100

120 ‘

100
120

FIG. 7. Intensity as a function of verti-
cal arrival angle determined by a 700-m
vertical array of hydrophones (Gaussian o
=180 m) centered on the sound-channel
axis (depth 1 km) at a range of 250 km

TRANSMISSION LOSS — dB

100
120

20 -20 0 20 -20 0
ARRIVAL ANGLE — deg

range, depth, and time. Results at 100 Hz show that
intensity fluctuations due to internal waves are signifi-
cant and comparable in size (5-30 dB) to those ob-
served in field experiments. Use of vertical beam-
formers as detectors has given insight into internal-
wave effects on the sound energy, and will probably
lead to a sensitive probe of the internal-wave spectrum,
In addition, selection of vertical arrival angle by use

20 km

60 km

100 km

from the CW source.

20 -20 0 20

of a beamformer significantly reduces fluctuations over
the single-hydrophone result.

We have presented only a small amount of data avail-
able from our computer simulation. In the future we
expect to present results on phase fluctuations, fre-
quency spectra of fluctuations, sensitivity to internal-
wave parameters, and comparison with simpler calcu-

250 km

—10°

=
fo)
co)

FIG. 8. Ray-peak transmission-loss
time series at several ranges and
depths. Each section shows the inten-
sity that would have been observed by
a 700-m vertical array (Gaussian o
=180 m) looking at the ray peak. The
sections correspond exactly to those

TRANSMISSION LOSS — dB
foe)
ro)

shown in Fig. 6, The higher intensi-
ties in Fig. 8 are due to the summa-
tion over many hydrophones in the

nae vertical array. Note the lower fluc-
100 RA tuations in the single ray observations
here compared to the single hydro-
phone (multiple ray) observations
shown in Fig. 6.
80
+5°
100
~l 1 1 1 1 Ne a ale
50 100 50 100 50 100 50 100
TIME —h

207

SINGLE
HYDROPHONE

rms INTENSITY FLUCTUATIONS—dB

AXIS
RAY

+5°

0 100 200
RANGE—km

FIG. 9, Intensity fluctuations as a function of range for sever-
al rays. The solid line indicates the fluctuations from a single
hydrophone placed at various ranges along the way. The rapid
oscillations in the solid line are due to the rapidly changing
multipath environment, It is interesting to note that the rms
fluctuation from a large number of paths with random phases
is expected to be 5.6 dB. ‘9 The dashed line indicates the fluc-
tuations observed in the ray peak determined from a 700-m
vertical array (Gaussian g=180 m). The result of selecting a
single path is seen to be a reduction in fluctuations and a
smoother dependence of these fluctuations on range.

lations of internal-wave effects on acoustic transmis-
sion. ®
ACKNOWLEDGMENTS

This work is part of a larger study of the sources of
acoustic fluctuations in the ocean begun by Walter Munk,

whose seminal influence and continual encouragement
we gratefully acknowledge. Important conversations
were had with Roger Dashen, Kenneth Watson, and
Fredrik Zachariasen.

Our work was largely completed during the 1974
JASON Summer Study under the auspices of Stanford
Research Institute. The support for our work has come
from the Advanced Research Projects Agency, and part
from the Office of Naval Research.

APPENDIX A: VERTICAL BEAMFORMING

The code we are using propagates sound waves from
a point source along a vertical plane in an ocean with
internal waves. In order to determine the directional
character of the arriving signal at some position down-
range from the source, we have formed a vertical ar-
ray of receivers and combined the Signals with phase
delays to amplify the waves coming from particular
directions.

Suppose ¥(z;) are the wave amplitudes at a set of N
points at a particular range spaced equally in depth z.
The N points span the ocean depth z,,,, so that z,.,
= Nd where d is the spacing of the grid of receivers (d
is 15.6 m in our case),

We define the amplitude arriving from a particular
direction @ at a depth z as

o(0)= D¥edexp~ 3S

Zz ; :
= ) exp|[—ik,(z; -— z)sin@],
where o is a measure of the vertical aperture of the
Gaussian array and &, is the acoustic wavenumber.

We have chosen o=180 m so that the angular resolu-
tion of the array is 0.5° at 100 Hz and the expected in-
crease in intensity for a plane wave arrival, due to the
large number of hydrophones being summed, is 14.6
dB. Also note that sidelobes of the receiving array are
eliminated by the use of Gaussian shading—a practice
that is easy to implement in our numerical experiments
but inefficient in a field experiment.

‘nM, Ewing and J. L. Worzel, “Long-range Sound Transmis-
sion,” Geol. Soc. Am. Mem. 27, Part III (1948).

*See, e. g., P. R. Tatro and C, W. Spofford, “Engineering in
the Ocean Environment, ’ 1973 IEEE Int. Conf. , 206-216; J.
Northrop and J. G. Colborn, J. Geophys. Res. 79, 5633—
5641 (1974).

5See, e.g., R. H. Nichols and H. J. Young, J. Acoust. Soc.
Am. 48, 716 (1968); B. E. Parkins and G. R. Fox, IEEE
Trans. AU-19, 158 (1971); J. G. Clark and M. Kronengold,
J. Acoust. Soc. Am. 56, 1071-1083 (1974); G. E. Stanford,
J. Acoust. Soc. Am. 55, 968—977 (1974).

40, S. Lee, J. Acoust. Soc. Am. 33, 677 (1961); J. C. Beck-
erle, J. L. Wagar, and R. D, Worley, J. Acoust. Soc, Am.
44, 295 (1968); J. C. Beckerie, J. Acoust. Soc. Am. 45,
1050 (1969); E. J. Katz, J. Acoust. Soc. Am. 42, 83 (1967);
Vv. A. Polyanskaya, Akus. Zh. 20, 95 (1974).

5C. Garrett and W. H. Munk, Geophys. Fluid Dyn. 2, 225-264
(1972); C. Garrett and W. H. Munk, J. Geophys. Res. 80,
291 (1974); W. H. Munk, private communication (1974).

8F, Zachariasen and W. H. Munk, unpublished.

"S. M. Flatté and F. D. Tappert, “A Computer Code to Calcu-
late the Effect of Internal Waves on Acoustic Propagation in
the Ocean,” SRI publ, (in press). (Note that the internal-

208

wave spectrum described in this article is an updated, dif-
ferent spectrum from the one in the SRI publication, )

5H, A. DeFarrari, J. Acoust, Soc. Am. 56, 40—46 (1974).

§R. P. Porter, R. C. Spindel, and R. J. Jaffee, J. Acoust.
Soc. Am, 56, 1426-1436 (1974).

'0W. H. Munk, J. Acoust. Soc. Am. 55, 220-226 (1974).

6. M. Phillips, Dynamics of the Upper Ocean (Cambridge U.
P., Cambridge, England, 1966),

Mu, Milder, “Users Manual for the Computer Program
ZMODE,” RDA-TR-2701-001, R&D Associates, Santa Moni-
ca, CA (July 1973),

37. Leontovich and V. Fok, “Solution of the problem of prop-
agation of electromagnetic waves along the earth’s surface
by the parabolic equation method,” Zh, Eksp. Teor. Fiz,

16, 557 (1946),

‘“P. D. Tappert and R. H. Hardin, in “A Synopsis of the AESD
Workshop on Acoustic Modeling by Non Ray Techniques, 22—
25 May 1973, Washington, D. C.,’’? AESD TN-73-05, ONR,

Arlington, VA (Nov. 1973).

oe DYE Tappert, “Parabolic equation method in underwater
acoustics,” J. Acoust. Soc. Am. 55, S34 (A) (1974),

‘8k. H, Hardin and F, D, Tappert, SIAM Rey, (Chronicles)
15, 423 (1973); F. D. Tappert and R, H. Hardin, Proceed-
ings of the Eighth International Congress on Acoustics (Gold-
crest, London, 1974), Vol. Il, p. 452.

We have ignored the fact that the internal waves are moving
slightly while the acoustic signal is propagating (due to the
finite speed of sound), This approximation is justified since
the sound travels much more than a horizontal correlation
distance during a time short compared to the internal-wave
correlation time (Kenneth M, Watson, private communica-
tion, 1975).

'8We are indebted to H. Brock and C, W. Spofford of the
Acoustic Environmental Support Detachment, ONR, for pro-
viding this subroutine,

‘3 Dyer, J. Acoust. Soc. Am, 48, 337 (1970).

209

SOUND PROPAGATION THROUGH A FLUCTUATING
STRATIFIED OCEAN: THEORY AND OBSERVATION

W. H. Munk

F. Zachariasen

Reprinted from the Journal of the Acoustical Society of America
Vol. 597 NO 4;,.6010—-030,-.ApeLw 2976

Palla

Sound propagation through a fluctuating stratified ocean:
Theory and observation*

W. H. Munk

Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, California 92037

F. Zachariasen

California Institute of Technology, Pasadena, California 91109
(Received 18 November 1975)

We have derived expressions for the mean-square phase and intensity fluctuations and their spectra for cw
sound propagating through a channeled fluctuating ocean. The “‘supereikonal” approximation reduces to
the geometric optics (eikonal) limit for short acoustic wavelengths: A€27 L},/R and A<Lj,/(R tan’@), where
L, and Ly are horizontal and vertical correlation lengths of the fluctuations, R is range, and tan@ is the
ray slope, replacing the traditional (and much more severe) Fresnel condition A<27 L’/R for a
homogeneous isotropic ocean. The results can be expressed in closed form for an exponentially stratified
ocean model and associated “‘canonical sound channel,” with superimposed fluctuations from an internal
wave model spectrum based on oceanographic observations. The parameters are the stratification scale B,
the inertial and buoyancy frequencies wi, and n(z), the scale js of internal wave mode numbers, and the
internal wave energy per unit area. The results are in reasonable agreement with numerical experiments
based on the parabolic wave equation. For the “‘singlepath”’ 4-kHz transmission over Cobb Seamount the
observed and computed rms fluctuations in phase are 1.6 and 2.5 cycles, respectively; in intensity these are
5.5 and 2.2 dB, respectively, with anomalous intensities measured at high frequencies (“‘sporadic”’
multipathing?). For the multipath 406-Hz MIMI transmission, we obtain 4 10~* and 5 107? sec~',
respectively, for the experimentally determined and the computed rms phase rates.

Subject Classification: [43] 30.20, [43] 30.40; [43] 20.15.

LIST OF SYMBOLS

C(x), 6C(x) sound velocity and fluctuations

GE): 16 Geac mean velocity profile (z upwards); C at surface z =0 (ignoring mixed layer),
at channel axis Z=—h, at ray apex 2

rms (6C/C))=4.9%107* fractional surface fluctuation

n(z); No 2, 2 buoyancy (Brunt-Vdisdld) frequency

p(x, y) -(@ sc covariance of fractional fluctuations

V =2q?5C/C perturbation “potential”

0, q N=20/q frequency and wave number of sound signal; wavelength

w, k; Re =k? +k, k,=k, frequency and wave number of internal waves (H horizontal, V vertical)

p(x, t) expli(qx - ot)] sound pressure

XGTU log pressure, intensity, and phase

G(x), G(k) Green’s function and Fourier transform

ll perpendicular and parallel to ray path

V(rx, Ty) dimensions of ray tubes:

Jo Gis, 16.2) correlation lengths

n=(z-Z)/zB dimensionless distance above ray axis z2(=—1 km); B(=1 km) is stratification
scale

€=5.7x10° perturbation coefficient [Eq. (84)]

RAR Rak ERoe ranges of upper, lower and combined ray loops; axial ray loop

Oy) Sm ORR OV enor arc distances of upper, lower, and combined ray loops

6 =R* d®S*"/d(R*")? ray parameter [Eq. (86)]

6, 6 ray inclination, axial ray inclination

R radius of curvature of ray

212

Spreps

Jie Ge
ihe Eq. (90), usually j, =3
(71) =0. 435 Eq. (95) for j, =3
F(w,3;z), G(w), H(j) internal wave spectra
E,(w), E,(w)

Oea7e3xX100> secr
w? = 09, +n? tan’@ frequency limit
@=w/2n, n=n/2n, etc.

3,,=24 cph, %=3 cph, 7=1.10 cph

summation over ray path7, internal wave mode j, ray loop k

acoustic phase and intensity spectra

inertial frequency at 30° latitude

cyclical frequencies (usually in cycles per hour, cph)

A =ntané/w,, Eq. (99)
a, y, B=ay Eq. (107)
&(B; j,) Eq. (109)
Bae Bs ify ify Eqs. (96), (98), (102), (104)

INTRODUCTION

Sound scintillations in the sea may be regarded as the
result of weak scattering. The fluctuations in sound
velocity are small, typically 5C/C=5%x10" in the upper
layers, 3x10 at abyssal depths. But the range of
propagation can be very long, and the cumulative effect
pronounced. The fluctuations impose the ultimate limit
to the acoustic resolution of objects, similar to the
resolution limit of ground-based telescopes due to
“atmospheric seeing.”

Our purpose is to contribute toward a quantitative
connection between two observational programs that
have paid scant and reluctant attention to one another.
Measurements and analysis of the fluctuating sound
transmission have viewed the ocean as a transmission
channel and described its properties by certain corre-
lation functions that are not readily identified with known
physical processes. Oceanographers have studied
ocean variability with emphasis on the associated fluxes
of momentum, energy, salt, etc. Starting from an
idealized (but not absurd) model of ocean variability, our
goal is to compute certain quantities of experimental in-
terest, such as the mean square phase and intensity
fluctuations of received sound pressure, and to compare
these with measured values,

The procedure is to derive a formalism for which
the geometric optics limit valid for short ranges is
transparent, and the transition to larger ranges is easi-
ly visualized. Section I gives the solution for a homo-
geneous ocean, with the geometric optics limit subject
to a Fresnel condition [Eqs. (35) and (39)]. But the
real ocean is not homogeneous, nor is it isotropic. In
Secs, II and III the solutions are generalized to apply to
an inhomogeneous ocean [Eqs. (57) and (62)] and to the
special case of a vertically channeled ocean [Eqs. (66)
and (76)], respectively. The range of validity of geo-
metric optics is now actually extended, and in addition
we are able to obtain rather simple analytic expres-
sions for quantities of experimental interest both within

213

and beyond the geometric optics regime.

We next introduce a specific gross sound velocity
profile C(z) with perturbation 6C due to vertical strain-
ing of the gross structure from internal wave activity:

C(x, y, 2, t) = C(z)+6C(x, y,2, t)+- sacar

The effect of horizontal flow associated with internal
waves is smaller than that of vertical straining. We
also ignore intrusive and other forms of fine structure
which, apart from their intrinsic temporal evolution,
are carried around by currents and internal wave mo-
tion.

There are two immediate questions: are internal
waves and internal tides (in constast to turbulence,
planetary waves, ... ) the principal source of fluctua-
tions? Do we have an adequate statistical model of
internal wave activity?

To the first question the answer is yes within a fre-
quency interval between cycles per day and cycles per
hour. Vertical displacements by internal waves are
typically tens of meters and swamp other sources of
fluctuations. The oceanographically more important
planetary waves (related to the variability in ocean cur-
rents) are associated predominantly with horizontal dis-
placements and are therefore of less consequence to the
sound field; their frequency range is typically cycles
per month to cycles per week. But in special frontal
zones (e.g., near the Gulf Stream) the long-period
changes in sound transmission are probably planetary
wave related. Small-scale turbulence takes over below
the Richardson length’ (¢€/n’)!/?=order (1 m), But for
intermediary scales the buoyancy effects are predomi-
nant and the fluctuations are internal wave related.
Reliance on laboratory concepts of homogeneous iso-
tropic turbulence, though fashionable, seem to us to be
entirely misplaced.

The answer to the second question is no. We shall
apply the internal wave models by Garrett and Munk

(henceforth GM72 and GM75) with subsequent modifica-
tions.” The model is contrived, andnotthe result of the
kind of the three-dimensional array measurements that
are really needed. Information with respect to the high
wave-number cutoff is particularly lacking. Still, there
is sufficient evidence now to make the present exer-
cise rewarding. And where evidence is lacking it points
toward the inverse method of using acoustic observa-
tions to improve the description of internal waves.

We shall take an exponential stratification scale of
1 km. This same ocean model underlies both the in-
homogeneity of the sound field [e.g., the “canonical
sound channel C(z)’’] and the anisotropy of the 5C fluc-
tuations (the ratio of vertical to horizontal scale is
typically 1:10). The solutions are now in simple form
and require at most a single numerical integration;
even this can be avoided in most applications by evaluat-
ing the integral near the ray apex.

Objections will be raised to the application of a model
ocean. There are, of course, large geographic varia-
tions of the water column (acoustic experiments invari-
ably fall into “anomalous” regions), Our position is
that the geographic factor in sound transmission needs
to be (and is in fact) taken seriously; but what is even
more needed are explicit solutions that permit compari-
son with experiment and provide an insight into the role
played by various ocean parameters, provided the un-
derlying model, though idealized, has the fundamental
properties of the world oceans.

|. HOMOGENEOUS OCEAN
The problem is to evaluate the pressure
Re[p(x) exp(- iat)] (1)

at a point x produced by a point source at the origin of
frequency o and wave number gq (for convenience we
take the source to have unit strength). We then write

p = exp(iqx) (p; + ip) =| p| explid) = poe* ,

: (2)
X=X,+iX; ,

where po(x) = exp(iq! x! )/47| Xl = exp(iqx)/4nx is the re-
corded pressure amplitude in the absence of any fluc-
tuations, but allowing for geometric spreading. Hence
o=X, is phase, and

t=log,|p|?=(+)+2X,, (t)=log,| po (3)
is intensity (multiply by 10/log,10 to obtain dB; from
now on we write log for log,).

We assume the 6C fluctuations, and hence within the
approximation we shall use, the X fluctuations, to be
Gaussian, so that (X)=0, (e*)=exp(3(X*)). Some of
the interesting observables are given by

(ce?) = ((log| p|?)®) = (c)? + 4(X?) , (4

)
T=log (|p|?) =(e) +2(x2) , (5)
log| (p)|2 =u) + (x2) = (x2) , (6)
(2) =(x2), (be) =2(X,X), (7)

2
(?) = | Pol? [exp(2(X? )) + exp(2(x** ))

+2 exp(2(X;))], (8)

where

‘x2
(*1)=4|x]?)¢Re(x2)), (X,X,)=41Im(X*). (9)

Let us begin our discussion by neglecting the effects
of the sound channel. The problem of sound propagation
in the presence of fluctuations superimposed on a homo-
geneous isotropic background is easier to set up and to
visualize than the problem involving an inhomogeneous
background, so it is conceptually advantageous to work
out this case first. Later, when the inhomogeneous
background representing the soundchannel is introduced,
the analysis Can be carried out very much as in the
homogeneous case, and the resulting formulae, while
geometrically more complex, are entirely analogous
to those obtained in the simpler example.

Our analysis will be based on the supereikonal ap-
proximation, and it will be convenient at this point to
give a brief review of an earlier report of this method. °
The sound propagation from the source to the point x
is through an isotropic homogeneous ocean, in which
the sound speed is C, on which is superimposed a fluc-
tuation in sound speed 5C(x) which is, of course, very
small compared to C. Mathematically, then, the
pressure satisfies the wave equation

(V7 + q?)p(x) = VR)p(X) , il

where q=o/C for a source emitting sound of frequency
o, and where

V(x) =2¢ 6C(x)/C . (11)

{For the inhomogeneous case we shall simply replace
C by C(x), and, accordingly, g by ¢/C(x).] The bound-
ary condition associated with Eq. (10) is that as x—0,

p(x)—1/4rx. (12)

Equation (10) may be cast into integral form through
the use of the outgoing wave Green’s function

(v2 +g?) G(x -y) =55(x-Yy); (13)
explicitly, we have
G(x) = expligx)/4rx . (14)

Then we may write, in place of Eq. (10),
p®)=6@)+) aFCR-DVGpW - (15)

Iteration of this integral equation generates the pertur-
bation series for p(x), which is more conveniently writ-
ten in Fourier-transformed form as follows:

rs ae ma
ph) +6@\ Ge VEeR-k,)

37, 3 wa ee
: cm f an j on v(k,)G(k -k,)
x V(k,)Gk -k, -Kk,) +--+, (16)
where
G(k) =(F - Gi +iey , (17)

and, of course, where

214

p(R)=[ dRemp(-ik- DG) « (18)
The supereikonal approximation now consists of ne-
glecting all momentum transfer correlations in the per-

turbation series. That is, we approximate (k- k
—-k-+++—k,)’- qi tie by P—2k: (+ ko t-++ +h,)
+he+ +++. +h-q tie, and neglect all terms of the
form k, . k, when i#j. Note that the first approximation
occurs in the second-order term in V. Once this sim-
plification is made, the perturbation series can be
summed exactly, and one obtains the result

1/2 r2 a
p(x)= ce reer (eat gs +206, %) ie) , (19)
0

where

I(B, X) = jo @n)3 Eval ds exp[+i(sk + X+Bs(1—s)k?)] .

(20)
This expression constitutes the supereikonal approxima-
tion to the pressure. The conditions under which it is
valid are

qx>1, qL>1,

and

x <(1/q?L) (C/6C? (21)

(the last condition may in fact be too stringent).
Here L is the correlation length of the sound speed
fluctuations—i.e., the correlation function p(x — y)
=(V(x)V(¥)) /4q* vanishes when |x -y|2L

It is worth noting that if in Eq. (20) the Bs(1-s)¥
term is omitted from the exponent, we obtain

p®=7— exp [ie +f V(sx) as) "|

which is the conventional WKB, or eikonal, or geomet-
rical optics, approximation to the pressure. The pri-
mary virtue of the supereikonal form, therefore, is

that it contains as limiting cases both the conventional
eikonal and the complete first-order perturbation-theory
approximations.

(22)

While Eqs. (14) and (20) do constitute a closed-form
solution for the pressure, the expressions are still a bit
unwieldy, and further simplification is useful. To this
end, let us evaluate the integral in Eq. (19) by station-
ary phase, keeping in mind that x and q are both large.
The stationary phase point is By, where

a - ) =
2 . =
qd - ae z+ i(Box) + Bo ap 1(B, X) ery eet) p

From Eq. (20), we may estimate that
- og 6C
Bo 5g M6 ®)),, LC * Bo.
Hence, if
x<qL?C/6C , (23)

the stationary phase point is accurately given by the
solution of the simpler equation

G - 7/4 =

PRAUS)

and is located at By)=x/2q. Thus we find

hae exp(iqx)

re exp[(ix/2q) I(x/2q, x)] . (24)

To approximately evaluate the integral J in this expres-
sion, we return to Eq. (19) and now expand in powers
of the potential V in order to obtain the first-order con-
tribution to the pressure:

nt" frenllbe- ds)

x [iB log/(B, x)] .

We may evaluate this integral by stationary phase as
well; under the condition (23) the stationary phase point
is again at B)=x/2q, and we obtain, in analogy to (24),
the expression

+, expligx) [. x aa
py(x) = ree i 24 logs (> .3)| °

But we also know that

pi(x) = j oR G(x - x’) V&)GR’) .

Thus we may eliminate I(x/2q, x) from (24) to obtain

p(x) = G(X) explX(x)] , (25)
where
_+ Ser =e pet} ap
x@)=5¢ ) a Gk - x’) V(x)G(x’) . (26)

This is known as Rytov’s approximation to the pressure.
A direct derivation of it may be made by replacing the
wave equation (10) by an equation for log[ p(x)/G(x)] and
solving this to first order in V.* However the justifica-
tion for the approximation is somewhat obscure in this
direct derivation; in the approach via the supereikonal
technique what is being left out is more clearly visual-
ized.

In any event, depending on the validity of the criterion
[Eq. (23)], one may use either the superikonal expres-
sion [Eq. (19)] or the Rytov expression [Eqs. (25) and

(26)] to proceed further. We shall use Eqs. (25) and
(26).
The Green’s function G is given by Eq. (14); hence

we may write for the quantity X the expression

X(X)= 4 ~~ | ay eaT explig(y + |¥-¥F| - x)] VV)

(27)
Let us first comment on the geometrical optics limit
of this expression. This limit results from an evalua-
tion of X(X) by the method of stationary phase. Provided
that the Fresnel condition

x< ql?

is met, the stationary phase path in Eq. (27) is the
straight line joining 0 to X, and the stationary phase
value of X(x) is just

X(x, 0, 0) = FA ax! V(x’, 0,0) , (28)

which is immediately recognized as the correct geo-

metrical optics expression for the phase. The analogous
expression for the amplitude in geometrical optics is
obtained by keeping the second-order transverse deriva-
tives in V as well.

Returning to the general expression [Eq. (27)], let
us first evaluate (|X(X) 17). For convenience, we shall
choose X to lie along the x axis, so that X=(x,0,0). We
evidently have

1 2 va
Vie ale 3S d3¥. = => >
«| x(®) |?) GY a9, Ye WIR —F, vel —Te |

-¥2) expliq(y, +|®-J,| — x)|

x4q'p Gi
xexp[- igy2+|¥-Fe| - x], (29)
where we have introduced the correlation function
- Jo) =(V(¥,) V(¥e)) « (30)

We assume p to be independent of
of a homogeneous background.

4q‘p(¥,

3(¥,+¥2) for the case

It is convenient in Eq. (29) to shift to relative and
center-of-mass coordinates. We define
¥=¥i-Yo, ¥ =2(¥i 432) - (31)
Then, if we assume that p(¥) cuts off for values of y

2L, where L<x, we may expand in y/Y, Thus Eq.
(29) becomes

xp(y) exp\(igy)- [Y-(x- ¥)] , (32)

where Yand x- ¥ stand for unit vectors in the direction
of ¥ and x Ve respectively, and we have written
|\Y4391=Y and |k-Y+4¥| = |X-¥1 in the geometrical
factors multiplying the exponentials. This approxima-
tion introduces a negligible error.

The integral over d°Y may now be evaluated by sta-
tionary phase. The stationary phase path is the straight
line joining 0 to X, and the result is

(| XC) |?) = qPx f dy, p(y, , 0) ’ (33)

where y, refers to the component ¥ in the direction
parallel to x,

Introducing the Fourier transform of the correlation
function

pm) =| aFexp(—iK- How), (34)

permits us to rewrite Eq. (33) in the sometimes more
convenient form

(|X 2) = 23 { aR plo, K,) (35)

where “1” refers to the directions perpendicular to x.

= i a (X®), We now h instead of
X(x)|2)=4 j 3 a j at Next let us turn to . We now have, instead o
(| x05) |?) 4n? a YY —xl? u/ Eq. (29), the expression
ee ee ee ee |
x
2 3 = = : ss] _ ; o 2|_
(X(X)*) = pant d fa Ve 1X —F,1 y1%—Jel p(¥, —F2)explig(y, + |*¥-F, | x)} exp[ig(v.+|X-¥F2| —x)] . (36)

We again shift to the variables ¥ and ¥, and appeal to the vanishing of p(¥) for y2 L to justify expanding in y/Y and

y/|X-¥|. We obtain

4
(xt) )= gh [ a°Y se

As before, we may evaluate the integral over d°Y by stationary phase.

(x(k pi (" as { dy, ( d*¥ip Ov) jem |E(F+

“)

where, again, ” and “1” refer to directions parallel
and eat to x.

At this point it is convenient to express p(y,, ¥,) in
terms of its Fourier transform, as given by Eq. (34).
The integral over dy, d*¥, can then be carried out, and
we finally obtain the relatively simple expression

2y2\ _ ¢ Zr r Ps
(x) )-- Gf aK, (0, ral as

x exp[i(k?/q)(s — x) s/x] . (39)

Equations (35) and (39) constitute our central results.
They express the quantities of interest as integrals
along unperturbed ray paths (in this case straight lines)
of the Fourier transform of the correlation function
p(k) times rather simple geometrical factors. As we
shall see later, entirely parallel expressions obtain in

216

a expliq(¥+|2-¥| - x] | d 5y ply) exp if ( .

y- F- ¥P

1x-¥I

=1E-G-HIY) en
This yields
: ) sf], (38)

Mi eS:

—-SSS

the more difficult case of an inhomogeneous background
medium.

The expression for (| X(x)|®), Eq. (35), is precisely
the same result for this quantity obtained by using geo-
metrical optics to compute X(X) itself, and then calcu-
lating (| X(x)|?) from this. [This is easily seen by re-
ferring back to Eq. (28).] In contrast, Eq. (39) is not
what one obtains for ( X(x)*) from geometrical optics.
Geometrical optics for this quantity is recovered if one
expands the exponential in Eq. (39), a procedure that
evidently is valid only if

KE (s— x)s
q =x

<<]

Since k,~1/L and s, x-—s~x, this condition can be more
familiarly written

x<qL? ;

which we recognize as the Fresnel condition under which
the geometrical optics approximation for X(x) itself was
valid in the first place.

Thus Eq. (39) constitutes an improvement over geo-
metrical optics, while Eq. (35) coincides with geometri-
cal optics. Conversely, geometrical optics for ibe)
is valid out to a very large range, while geometrical
optics for (X*) is valid only within the range x< Glen

It is of interest to study Eq. (39) in the limit of very
long range. As x~~, the integral over ds can be ap-
proximately evaluated, and we find*

(X?(X) ) =[ig’p(0)/27] (y+ log4qx—im)~ilogr, (40)

where y=0.577... is Euler’s constant, while for small
x, satisfying the Fresnel condition, we have the geome-
tric optics limit

2 «12
(x40) =— ff a? 510, (x- Bee) (41)

Between these limits Eq. (39) provides a smooth transi-
tion. In contrast to Eq. (40) we have from Eq. (35) the
result

(|X|? )~x

for both large and small x.

Il. INHOMOGENEOUS OCEAN

Now let us turn to the effects of the sound channel.
That is, we must replace the nonfluctuating sound speed
C in the homogeneous case by a (specified) function of
position C(x).

The wave equation for the pressure, which is our
starting point, now becomes altered from Eq. (10) to
the equation

[v2 +4°(%)] p(X) = V (RAR), 9(%) =0/ CR) ,
still with the same boundary condition Eq. (12).

(42)

We must first study the nonfluctuating part of the
problem, to evaluate the Green’s function in the pres-
ence of the sound channel. This satisfies

[v? +4°(x)] G(X, ) = 6°(%-F).
Note that G is no longer a function only of ¥-¥ as it
was in the homogeneous case. We shall assume that
geometrical optics provides a good approximation to
the nonfluctuating sound channel problem. This means
that we can represent G(x, ¥) as a sum of contributions

from each ray joining x and y. To be specific, we may
write

(43)

n(& 9)
G(x, y) = 7 G; (x, y) ’ (44)
i=

where n(x, ¥) is the number of rays and G, is the contri-
bution of the ith ray. We have, in particular, for rays
joining the origin and x,
Z
G,(%, 0) = K, (%, odexr(i| as ql%(s))), eerie
0
(45)

where ds is an element of path length along the ray,
%,(s) is the ith ray joining 0 to X, and K, is a normaliza-
tion factor.

Now when the fluctuations are turned on, the signals
traveling on each of the rays joining the origin to the
point of observation X are subject to small-angle scat-
terings by the perturbing potential V(x). The signals
are thus deflected slightly from the undisturbed rays by
each interaction with V. The repeated action of V thus
produces, on each ray, a sort of random walk of the
signal away from the original ray. When we average
over an ensemble of perturbations V, the disturbed sig-
nals will fill up a tube surrounding the undisturbed ray.
Provided that these tubes around each of the original
rays do not overlap, the received pressure will be a
sum of contributions from each ray tube, (Such tubes
exist, of course, in the homogeneous case as well, but
there they never overlap. )

We may estimate the radius of a ray tube as follows.
The mean free path d between interactions of the signal
traveling along a given ray with the perturbing potential
V is of the order of 6C/kC. Hence over a range x the
number of scatterings is n=x/d. The average deflec-
tion angle due to each scattering is of order 1/kL, ver-
tically and 1/kL, horizontally, where Ly and L, are the
vertical and horizontal correlation lengths of the sound
speed fluctuations. Since the process is a random
walk, the net dispacement due to n collisions (when n is
large) is proportional to vn, and hence the vertical and
horizontal extents of the tube are, roughly,

_ (xv? oe ee a ae
ng G) qly \6C} 7° * ay ql, \6C/ *

Let us assume that the vertical extent of the tubes is
small enough so that the tubes remain distinct. Then

the pressure at x is the sum of contributions from each
tube

(46)

where n(x) is the number of unperturbed rays joining the
source to the point X. We shall be interested in p,(X).

We note that p,(X) is the pressure that would be re-
ceived at X if the source were not isotropic, but rather
emitted all its energy in the direction of the 7th unper-
turbed ray. Thus p,(x) must satisfy the wave equation
(42) but with an anisotropic boundary condition which it-
self depends on X. To make this more precise, let us
define p, (¥; X) to be the pressure at ¥ from a source at
the origin which emits only within a small solid angle, 5
around the direction of the ith perturbed ray joining the
origin to X. Thus p,(x)=p,(x; xX), and furthermore
p;(¥; X) vanishes unless j¥ is inside the ith ray tube.
Then

[ve+q7°H)]oi Fs = VHF; *), t=1,..., 2%). (47)

In analogy with this definition of p,(¥; Xx), we may also
define an “unperturbed” Green’s function G,(¥; x, 0),
i=1,...,n(x), to satisfy

2187)

[v3 +42] C.F %, 0)=0 if F40,
(48)
G(%, 0) = 2 G,(% %, 0),
1

again with the same boundary condition. This function,
also, will vanish except when ¥ is near ith unperturbed
ray.

We may now directly derive the analog of Eq. (25) by
computing the quantity logp,(¥; X)/G,(¥; X, 0) in pertur-
bation theory, and using Eq. (43). We find, setting ¥
=k, that

pi(®)= 6,8, 0) ex (Se. x, on a!
x G(x, y) VIV)G, (9; *,0)) - (49)

Finally, we note that the presence of the function

G,(¥; X,0) in Eq. (49) will restrict the integral over d°¥
to a region surrounding the ith ray from 0 to x; since
inside this ray tube G=G,, we can write

Ee 1
pi(®) = G, (&, 0) exp (;

——_— Gey.
i(X, Thee tube

x GR HVGHCG, 0)) . (50)

We have here replaced G,(x; x, 0) simply by G,(x, 0).
Equation (50) is evidently the generalization of the Rytov
formula (25) to the situation of an inhomogeneous back-
ground and many rays. The expression clearly fails if
the range is so large that the ray tubes overlap; other-
wise the validity conditions are the same as those in the
homogeneous background case.

We shall now use Eq. (50) to calculate the various
averages of interest for the contribution of a single ray
tube to the pressure in the presence of the sound chan-
nel. We shall, for simplicity, drop the index i, though
we should keep in mind that when there are several un-
perturbed ray paths their contributions are to be added
to obtain the total pressure, Our interest, then, will

ee eee ee ee

(

K(x, 0)

and we must keep in mind that we are to integrate only.
over the ray tube surrounding the unperturbed ray of in-
terest. In the homogeneous background case we ex-
panded the exponent in powers of y, because p(y) van-
ished for large ly]. We may do the same here. Thus

(

x@)|*)=4¢" | °F SE DEE ON as5og, ¥)

xexp(iq{¥-V,[s&, ¥)+S(¥, 0)]}). (56)

The integral on d*¥ is again to be evaluated by station-
ary phase. The stationary phase path is evidently the
unperturbed ray joining the origin to the observation
point X. Hence we may write

be in the statistical fluctuations of the contributions of
a single ray, or rather a single ray tube.

As in the homogeneous case, we define
wos RS eS EES
XD-E__ | CICRNMHCG, 0), (51)

and we wish to compute (.X®) and (|. X|®). We recall that
assuming geometrical optics tobe a valid approximation
for the nonfluctuating background permits us to write
the Green’s function

K(X, 9) expligs(&, 9] , (52)
where
ast, D=(" ds afX€(s)1, (53)

and where the normalization factor is

1/2
) . (64)
%=¥,=0

Here 1 refers to directions perpendicular to the ray.
[An excellent approximation to this, for our purposes,
is to write simply K(x, y)=(471X-y|)", as in the homo-
geneous ocean case; we need to be careful about devia-
tions from homogeneity only in the phase.] In Eq. (53)
the line integral is along the ray of interest joining the
points X and y,

- > 1 8 te)
KR H)=g- (aet ==

e am, ay S(x, y)

To repeat our earlier calculations requires us to in-
troduce the correlation function p(¥,, ¥2). In the homo-
geneous case, this quantity depended only on the separa-
tion Y=y¥,;—Y2. Now, however, because of the back-
ground inhomogeneities, it will also depend on ¥
=4(¥,;+Yz) (actually it will depend only on the mean depth
3(z, + 22) because the inhomogeneities depend only on
depth). Thus we must now define the correlation func-
tion by

4q* ply, Y)=(V(¥1)V(V2)) .
As before, let us look first at (|X1°). We have

= ae Y z = ap ete a 1> ma Tics = >
xG)|*)=4¢' {ae (“SEO j d'¥ p(y, Y)exp{iq[S&, Y+ zy)- Sik, Y- 29) + S(¥+ 2y, 0) - S(Y-3Y, 0)]}, (55)

ee
(|x@)|*)= [asf a°R, 6) HRs), Fis), 7

in complete parallel to the homogeneous case. Here
the line integral on ds is along the unperturbed ray,

k, (s) refers to the component of K perpendicular to the
ray at s, Y(s) is a point on the ray at s, and

p(k, %) =( dy exp(—- ik.) p(y, Y) . (58)

Next we turn to (X*):

K(&, Y) K(¥, 0)
K(x, 0)

xexp {ig S(X, ¥ +3) + S(X, ¥ - 29)

(X(x)?) =4q4( a’¥ j d°¥ ply, ¥)

218

+ S(¥ +44, 0) +S(¥ —4y, 0) - 25(x, 0)]} . (59)

Now when we expand the exponent in powers of y the lin-
ear terms vanish, so that we have

3 KG, K(¥, 0)
(x@P)=4g"f a°¥ Get,
x exp{2ig[ S&x, ¥) + S(¥, 0) - S(X, 0)),
xf dF 0G, DexpltianyAyH], (60)
where we define
A, (¥) = S(x, ¥)+S(¥, 0)] . (61)
qf ae ay|s ]

Evaluation of the integral on a’y by stationary phase

again selects as the stationary phase path the - unper -
turbed ray joining 0 to xX. The integral on d*y can then
be done by introducing the Fourier transform p(k, Y) as
in Eq. (58). Finally we obtain

(x@)=-G fj as a%% (6) ARIF)
0

x exp (i/q)ki, (s)ki,(s)A*(¥(s)),5] « (62)

The notation is as in Eq. (57), and the result is again
in complete analogy to the homogeneous case,

Most of the comments we made in Sec. I concerning
the results in the homogeneous background apply here
as well. The expression for (|.X|*) is again just that
obtained in the geometrical optics approximation, but
that for (X®) is not. Geometrical optics for (X*) is
valid provided that

(1/q),(s)ki ,(s)A

This is the analog of the Fresnel condition.

(¥(s))yy «1. (63)

Il. CHANNELED OCEAN

Let us next apply our general results, Eqs. (57) and
(62), to the specific case of a channeled ocean with its
associated cylindrical symmetry. We shall choose the
z axis as the vertical with z positive upward. We shall
choose the unperturbed ray of interest to lie in the x, z
plane, and we shall confine ourselves to situations in
which the source and receiver lie at the same depth.
Thus the source is at the point (x, y, z)=(0, 0, 0) and the
receiver is at the point (R, 0, 0) where R is the range.
The unperturbed ray path joining these two points will
be denoted z(x); thus z(0)=0, z(R)=0 and

(x) =tan™ i)

is the angle the ray makes with the horizontal at the
point x. The element of path length along the ray is
then given by

ds =([1+tan@(x)]}/? dx.

The expressions (57) and (60) for (|X17) and (x?)
both involve integrals of the Fourier components of the
correlation function over a plane perpendicular to the
ray path at each point along the ray. With our geome-
try, the wave number K(s) perpendicular to the ray has

825

x, y, and z components
K(s) =(-

and the element of surface area perpendicular to the
ray is

k,tan®, k,, Re),

ak, (s)
Thus Eq.

= dk, dk, /Cosé .

(57) becomes
R co) cd
(ixl)=% j dx sec’@ i as, dk,
0 Ls oe

xp([—- k, tan(x), ky, Re], 2(x)) « (64)

Now it turns out to be more convenient to express the
correlation function p in terms of the variables w and j,
the frequency and mode number of the internal waves,
rather than the wave numbers k, and k,. The transfor-
mation to these new coordinates is accomplished as fol-
lows: horizontal and vertical components of wave num-
ber

=(Ktan?o+#)/? ,
have the approximate dispersion relations [Eq. (92)]
V8 /n,

Then we note the definition

ky =k,

ky =j7B(w? - w? ky =j7B™ n/n «

ack == 1 3
J Gp 2)= 2 [ha dln f de Ble he 2)

“Lf ta Ploy jz), (65)

where F(w, j) is the eater of 6C/C, and
we =we, tn’ tan’.

The internal wave vector kK has an inclination ky / Ry;
group velocity is at right angles, with inclination ky/ky,
and its component in the plane of sound propagation is
kh, /ky< ky / Ry. At w=wz,, ky /ky = we —wi,)'/?/n=tand;
lesser frequencies have k,/ky<tan@, and are excluded
from consideration in the stationary phase approxima-
tion.

Thus Eq. (64) may be replaced by
R n
(|x|?) = 2 "¢PngB { dxsec?@), nf dw
0 J w

L
X (w? = w2)t/? Fw, jj 2) « (66)

In this expression z, n,-and 6 are of course functions of
x, to be evaluated at each point along the unperturbed
ray as x varies from source to receiver.

A similar transformation may be carried out for Gey
the other quantity of interest. The only complication
here is that it is necessary to evaluate the matrix A,,,
introduced in Eq. (61), at each point along the ray.

The symmetry associated with a channeled ocean makes
this relatively easy to do, as follows. First, in the
horizontal plane the unperturbed sound speed is con-
stant. Hence the optical path between any two points
(19121) and (xpy2Z2) can be written

= y1)?/2(xp - ™)

+ S'(xe2e, 121) ) (67)

S(x2V2223 X19121) = x2 —

219

provided that x,- x, >v2—-1, Z2-2,. Therefore the
matrix A,; has the form

Ay. 0 Ay,
Ay=| 0 R/x(R-x) 0 ; (68)
Ae 0 A

xz ae

at an intermediate point x along the ray joining source
to receiver, where the 2x2 matrix Aj, now is defined in
the vertical xz plane only.

The matrix Aj, has the form

tan’?@ —tané
sae ae),
tandé 1

it has one zero eigenvalue, associated with an eigen-
vector parallel to the ray, and one nonzero eigenvalue
A! with eigenvector k, =(-k,tan@, k,) perpendicular to
vn ray. The quantity hykiy(A'*),, is just R/Aj. Since
=(1+tan’@)Aj,, we have B/A/=#/A,,=K/A,,
= a k, bre Thus computing any matrix element of AVG
is sufficient to allow us to obtain the quantity we need.
At this point we shall drop the primes.

There are a few regimes where Aj) may be obtained
without specific assumptions regarding the form of the
sound channel.

First, for very deep rays, the sound channel varies
nearly linearly with depth, and the rays are nearly arcs
of circles. For this case, to first order in the radius
of curvature, the quantity A,, has the same value as in
a homogeneous ocean:

Aji=x(R-x)/R.

Second, for near axis rays, the sound channel is
nearly a parabolic function of depth, so that the rays
are nearly sinusoidal. Then we find

Aj = (1/K) sinKx sinK(R — x)/sinKR ,

where 27/K is the wavelength of the sinusoidal ray.
Thus 7/K~R, the range of an axis loop. Note that this
approximation to Aj} becomes infinite when the receiver
is located at axis crossings of the ray. These points
are caustics for sinusoidal rays. Caustics are, in gen-
eral, points at which Aj} diverges, that is, where A,,
=0. When this occurs, the matrix A,, takes the simple
form
0 0 0
Ay=| 0 R/x(R-x) 0 5

0 0 0
so that the passage from Eq. (60) to Eq. (62) becomes

altered, because the second-order term in the trans-

eee See ee eee

verse derivative of the optical path no longer dominates.
To calculate (X*) correctly in this region, we would
have to keep anaes derivatives of S(&, Y) +S(Y, 0)
[cf. Eqs. (59)—(61)]. As we shall see in Sec. VIII, the
effect of caustics on our theoretical predictions is to
introduce false narrow spikes in (X*) at the caustic
positions, Presumably, there is in fact some unusual
structure at these points due to the different behavior
of the integral in Eq. (59), but our theory, to the level
to which we have carried it, cannot correctly describe
this structure,

A third regime in which we can obtain Aj! without
knowledge of the details of the sound channel is that of
very long ranges, in which the rays contain a large num-
ber of loops. In this case, the optical path length S, in
the vertical plane from the origin to a point (x, z), plus
the optical path length S, from (x, z) to the receiver at
(R, 0), can be written

S=S,((x, z), (0, 0)) + S.((R, 0), (x, 2))
See peel Poa ee z), (69)

where m;,. are the number of double loops in the first
(second) path, Sf”, is the optical path length. of one
double loop, Rj, is the range of one double loop, and
AS, 2(x, z) is the remaining path length from the end of
the last double loop to the point (x, z). Evidently

x=mRy+4x,, R-—x=n,Rz +Ax ,

with 4x, ,.<«<x, R-xand 4S, ,<Sj',. Thus we may ap-

proximately write

S=n Sf (x/m) +ngSz(R— x/nz) . (70)
Therefore

BE:

Since n,+n.=R/R*, and since n, =x/R*; n.=(R—-x)/R*,

we have
«x = [R/x(R—- x)]65 ,

EL Nida S naa)
are

unperturbed ray ‘ Ne

6=R*d’s*/d(R*) , (72)
and therefore
Az, = (tan?6/5) x(R—x)/R.

Other than in these cases, Aj} depends on the sound
channel, and we shall defer further discussion of it to
Sec. IV.

Let us now return to Eq. (62). Once we have obtained
Ags we can write the exponent in the integrand of Eq.

7 Ru (s)Ruy (SA 1% (ds =2 (e HB) (73)

R A

Hence, using the dispersion relations for horizontal and
vertical components of wave numbers, we find

R 2 2 2
(x) =2g4aIngB | dxsecté 51" du (w* = wt)? ? Flu, j; ayexr| | (2) é e x) w oa 3 )| : (74)
0 ty wr No No gg

Geometrical optics is valid when the exponent in this
equation is much less than one; this is the analogue
of the Fresnel condition in the channeled ocean.

ee
Except when the receiver is in the vicinity of a caus-

tic, it is in general the case that the ke term in the ex-

ponent, which is associated with horizontal spreading,

220

is much smaller than the term with Aj. This is be-
cause, as we shall see below, the spectrum F(w, j; z)
tends to weigh small values of w much more heavily.
Thus for most purposes we can ignore this term, and
replace Eq. (74) by the much simpler expression

R
(X®) =— 2q?n ngB { dxsec’6 me - exp (ipj*)
0 uy

x dw

F(w, Jj; 2)
(w*- wy) /*’ 9)

where
B=(1/B)* (n/n) 1/gA,, (76)

Using the special expressions for A,, derived above ap-
plicable to specific regimes; we can write, for single
loop downward rays,

pera SY ge

for near axis rays

Aa TY (/n¥ 1 sinkxesinK(R- x) .
ACB) ON in) KG sinKR ;

and for the long-range many- loop situation,

_({1rV¥ (ni x(R- x) tan?6
p=Pm= (3) le ) Rye aGi 7
0 q

Equations (66) and (75) are as far as we can go toward
computing the quantities of interest without committing
ourselves to a particular spectrum F, and a particular
sound channel and associated ray paths. It therefore
now becomes necessary to turn to a discussion of these,
and Secs. IV-VI will be devoted thereto,

IV. CANONICAL SOUND CHANNEL

Let T(z), S(z), P(z) designate the undisturbed distri-
butions of temperature, salinity, and pressure. The
velocity of sound is a known empirical function of these
variables, C(z)=C(T7, S, P), having typically a minimum
value C=C at some depth z=— hand increasing by a few
percent towards top and bottom. The (fractional) veloc-
ity gradient can be written

C10,C=a+ ,T+B- 8,S+y: 9,P, (77)
with
(a, B, y)=C™+ (87, 25, Ap)C.

The temperature gradient is the sum of potential and
adiabatic gradients, 6,T=9,Tp+9,T,, so that

C18,C,=a+ 8,7, +7 8,P=(-0.03-1,.11)x10% km™
=-1,14x10"% km?=-y, (78)

is the fractional velocity gradient in an adiabatic iso-
haline ocean. In analogy, we define a potential velocity
gradient such that

a,C = yp Ts a,C, = 8,Cp a Cr, ’

and write the potential gradient in terms of the buoyancy
frequency n(z):

C71a,C= (u/g)n?(z)— v4 , (79)

n°(z)=— gp 8, pp = g(a9,Tp — b8,S) = ga2,Tp(1 — Tu) ,

(80)
where the “Turner number
ba_S
pay dS? tana
oe ao,Tp

gives the relative contributions of salt and (potential)
temperature to the potential density stratification. In
Eq. (79)

u=(a/a)s(Tu), a/a=24.5,

s(Tu)=(1+cTu)/(l— Tu), c=aB/ab=0.049 ,

using the numerical values
O=3. 19x10 (Cr,
B=0.96X107* (%)7 ,

y=1.11X10 km.

a=0.13x10°5 (°c)? ,
b=0. 80107 (%)7 ,

The a value is typical of conditions in the sound channel
(it may vary by as much as 50% between surface and
bottom). In shallow water the n? term dominates, and
the velocity increases upwards; in deep water n?—0 and

the velocity increases downwards at the rate y,. At the
axis of the sound channel 6,C=0, hence
n(z=-h)=n=(gy,/p)”? . (81)
An exponential stratification model
n=ne/?, B=1km,
(82)

ng=5.2X10° sec? (= 3 cph),

gives a reasonable fit to the oceans beneath the thermo-
cline® (we ignore the surface mixed layer and interpret

No as a surface extrapolated value). The sound axis is

at a depth

— Z=h= Blog(no/n)=0.89 km+4Blogs , (83)

compared to typically observed values 0. 7—1.5 km.
Geographic variations in the sound axis are associated
with the temperature dependence of » through the a pa-
rameter, and with the salinity dependence of s(Tu). We
take h=1 km. In terms of a dimensionless distance 7
above the sound axis, the velocity profile can be simply
written

C=C[1l+e(e"=n-1)], €=} By, =5.7x105,
n=(z-2Z)/zB.

The coefficient € is readily interpreted as the fractional
adiabatic velocity increase over a scale depth. Equa-
tion (84) is a reasonable description of an oceanic sound
channel (Fig. 1), given in terms of physical constants
of seawater and the stratification parameters no, B, Tu.
We require certain geometric properties of the sound
channel. Let z(x) denote a ray with inclination dz/dx
=tan@ and curvature d?z/dx =sec’@: dé/dx. From
Snell’s law cosé = C/G: where C is the velocity at the

(84)

ray apex. Hence

ppndnz dé 2€C

1 = Vo sabes ——— — 7) -e"
R Sage oe () ae tan@ Be sec6(1- e")=y,(1-e") .

(85)
The range of a double loop® is

km/sec

fe) 1.50 1.52 1.54
1
= 2
xs
=
5
le
+
(ou
o
O 4
5

FIG, 1. Canonical sound channel (left) and the corresponding rays for 6 =12.7° (surface limited), 5.2°, and 0° (axial ray). The
contribution to F; from various parts along the three ray paths is indicated by the vertical extent of the shaded band (plotted loga-
rithmically). Fy, is plotted separately at the bottom of the figure for the surface limited ray, together with F, and F;—ReG,, thus
indicating the relative apex contributions toward mean-square phase, rate of phase, and intensity, (F,—ReG, applies only toa

source at x=0 of a receiver at R*.)

RY =Rt+R =7Bet/? (1+4¢7+---) :

where ¢?=(C-—C)/(eC), and the “optical” path length
equals

Sah den Ben’?2G (1 +heb24. ere) ie
We will require
dest

6=R™ pie =~ 1e(L ++). (86)

Finally, we can (laboriously) compute Aj} for a com-
plete loop. We find
At 7 3B 1 ¢? ( & ¥2- oY
MGA 2e 6 3 ¢

2 $°(¢? - $*) @ 41 ¢)
a5 oe (\-5%5 sin 5) |

It will also be of interest to have the value of this quan-
tity at the apex of a ray. This is

Fees) al al g? V2 5)

1 = —_

Au=q ie rates 3 9) -

For upward rays, Aj}~xnear x=0 and Aj) = R*— x near
x=R* (Fig. 2), and Aj has zeros at the caustics of rays
propagating to the right from x=0 and to the left from

222

1.0 km

FIG, 2, Aj) for a 5, 2° upward loop (top) and downward loop
(bottom), with the +5, 2° ray itself shown in the center, all
plotted as functions of horizontal distance x,

x=R*. For downward rays, ie is reasonably well ap-
proximated simply by x(R—-x)/R.

V. FLUCTUATIONS IN SOUND VELOCITY

In the presence of an internal wave field with vertical
displacements ¢, a particle momentarily at z comes
from a rest height z—¢. The resulting velocity fluc-
tuation at a fixed depth z is

6C/C=5C/C=a5T+B5S+y5P ,
where

5T=-98,T,, 69=£-9,S, 5P=pgt- Ap/p

are the internal wave-produced fluctuations in T, S, P.
The wave-induced pressure fluctuations at a fixed depth
are reduced by a factor Ap/p(~10°8) over the pressure
fluctuation pgé experienced by a fixed water particle.
Henceforth the effect on 6C of the internal wave associ-
ated pressure fluctuations will be neglected, and so

8C=£(8,C—8,C,)=£8,Cp ; (87)

e.g., the internal waves convect the potential velocity
gradient as defined in Eq. (79):

6C/C=n"(z)ut/g. (88)

The rms vertical displacement is given by (GM72)

“1/2

rms(¢)=rms(£q)(n/n9) » rms(f))=7.3m,

relative to its near-surface value, and so increases

with depth as n7™\/?; accordingly

rms(6C/C)=rms(5C/C)9(n/n)/? ,
with (89)
rms(5C/C),= unsrms(f,)/z ,

decreases exponentially with depth with a scale 7B
=0.67 km. For orientation, set s=1 and u=24.5; typi-
cal values are given in Table I. In very deep water
5C/C is of order 10°, and accordingly the rms fluctua-
tions in sound velocity are a few cm/sec. The rms
horizontal velocity components associated with internal
waves are (GM72)
rms(u)=rms(u9)(2/n9) , ug =4.7 cm/sec ,
leading to the values in the last column. The last two
columns give relative perturbations in sound propaga-
tion associated with vertical displacement and horizon-
tal particle velocity, respectively. The latter effect is
much smaller (except in very deep water), and will be
ignored subsequently. On the other hand, the w effects
dominate at and below inertial frequencies, so that
planetary waves with their quasihorizontal particle mo-
tions affect sound transmission by Mach refraction.

TABLE I. Typical values with s=1 and w=24,5,

2 n n rms ¢

(km) (cph) (rad/sec) (m) rms 6C/C_ rms u/C
thermocline 2z)=0 3200, 5;2x10% 753) “4y9xio © “3.1xi07
sound axis Z=-1.9 1,10 1.9x1075 12.0 1.1104 1,1*107%
bottom z=-4.5 0,094 1.7x104 41.2 2,8x10% 1,0x10-6

Te Oi. —-s# lO ———WooOowuOWDaan—

Vi. INTERNAL WAVE MODEL

Fluctuations in the vertical structure of temperature
and salinity were discovered by Petterson, Helland-
Hansen, and Nansen soon after the turn of the century.
Since that time there has been a vast literature on the
subject (over 500 references were compiled by Roberts’)
consisting mostly of reports on temperature and current
fluctuations at moored instruments, and of a few hori-
zontal temperature profiles from tows behind ships. In
the past three years, the technology of continuous ver-
tical profiling of currents with freely dropped instru-
ments has been developed, providing additional infor-
mation. A three-dimensional trimooring (IWEX) was
installed in 1973 off the American east coast, and we
may expect some very useful additional results.

On the basis of this myriad of observations, Garrett
and Munk have contrived successive models’ (GM72,
GM75) of internal wave spectra. They placed particular
emphasis on multiple recordings, separated vertically
on the same mooring or horizontally on neighboring
moorings, which had shown that fluctuations of frequen-
cies as low as 1 cph were uncorrelated for vertical
separations exceeding a few hundred meters, and for
horizontal separations exceeding a few kilometers.
These coherences were interpreted as a measure of
reciprocal bandwidth: for separations larger than the
reciprocal bandwidth, different wave numbers interfere
destructively, and coherence is lost. The following con-
clusions were reached: (i) Observations can be recon-
ciled with the dispersion law and wave functions of lin-
ear internal wave theory. (ii) Towed records are in-
sensitive to the ship’s course, and moored records are
similar for the two velocity components, thus indicating
some degree of horizontal isotropy; the evidence is
certainly incompatible with internal waves propagating
along narrow horizontal beams. (iii) Coherences are
incompatible with a model consisting of just the gravest
one or two vertical modes (except at tidal frequencies).
The GM72 model had equal contributions from modes 1
to 20, and none beyond mode 20, But this is too broad;
recent measurements by Cairns® are consistent with a
mode weighting according to (j*+j%)* with j, ~3.

(iv) The myriad of observations, taken over the years
at many depths off the American west and east coasts,
Hawaii, near Bermuda and Gibraltar, in the Bay of
Biscay, and the Mediterranean, agree to within an or-
der of magnitude. This suggests some universality in
the internal wave spectrum, perhaps due to saturation
effects such as those limiting surface waves of high
frequency.

We use the GM75 spectrum, somewhat modified for
the Cairns observations:

na =
(2%(2)) =f “do Feluni2)

Fylw, j; 2) =(¢7(z)) Gw) HCY)

4 2_ 2 \1/2 nie)
GWw) == yal" = wi) , Glw) dw=1,
1 w Be

2233

Hi) = (+81 2+,

SE AG)=1;,
at

between the inertial frequency w,,=2 sin(latitude) cpd
and the buoyancy frequency n(z)>>w,,, and zero other-
wise, where

Do (+H) eh IP (nie 1) for j,21.
1
Similarly for the spectrum of 6C/C
Fe jo (w, 5; 2) = ((6C/C)?) G(w)H( 5)
=((6C/C)%) (n/no)® Gw) Hj) .

For w not too close to n(z), the dispersion relations

fo ee ee ee

(91)

§Sj F(k) cosk,xdk = a awl Gly FUbq » w) (20) da cos(ky cosaX) -{

in oa

n(z)

are

Rey = jn BB ngt(w? — w2,)'/? , ky (z)=jnB'n(z)/no (92)

for the horizontal and vertical components of wave num-
ber. The spectrum in k,,7 space is accordingly

: ., dw
Fye po (ay 9) = Fee sc (w 3) dky

acy’ 4(w4, /no) BT Kj H(3)
= = 93
(CC) etGetoc mre 9)
for 0< ky< ky =j7B'n(z)/ng. Equations (90)-(93) are
essentially WKB approximations, and they fail near the
boundaries and the turning depths (GM72).

Coherence scales can be estimated from the Fourier
transforms:

n(g)

dw | dky F(Ry, w)Jo(Ryx)
a 0

=| dws Glo) H(j)Jo(yx) ~1 — 81747, (wy, /no) [log n/ws,) - 3] | x|/B ,

yy H(j) coskyz =1- (nj, - 1) (n/np)|2|/B,
which suggest the coherence scales

es (no /w4,)B L _Bno/n
* 8j,{log(n/wy,)-2]? ” }

Te
Setting w,,=7.3x10"° sec” (30° latitude), m)=5.3x10%
sec, B=1 km, gives the values in Table II. (The
near-bottom value of Ly is meaningless.) The assump-
tion of spherical symmetry (so popular to scattering

theoriests) is useless to oceanographic application.

(94)

Vil. FLUCTUATIONS IN MODEL OCEAN

Armed with the specific ocean model described in
Secs. IV-VI, we can now proceed with the evaluations
of the general expressions (66) and (75) for the quanti-
ties (|X|?) and (X?), Let us first look at (|X1?).

Substitution of the spectrum (91) into (66) yields
(|X|?) =((6C/O)) (77)? BRF,(R) ,

8 w,, 1 ie 5 Me dw (S25 ee
= 6
Z a R pee cce n bona ust ae

(95)

F,(R)

TABLE II, Typical values for w,,=7. 3x10 sec (30° Lattitude),

ny=5.2%107% sect, and B=1km, The near-bottom value of
Iy is meaningless.

z n Ly Ly

(km) (rad/sec) (km) (km)
thermocline -0.1 5.210% Pb} 0.12
sound axis -1.3 1.9x10% 3.4 0, 32
bottom —4,5 1.7105" 27.0 (3, 63)

Ase lee ones
esa dxsec?6 n5f,(A) , (96)
T WiaNo R Jo
and
1 a? (A? +1)'/? +1
AA)“ BeT aR Tele (yi (ON)
with
A=(n/w,,)tané . (98)

F,(R) as here defined is a dimensionless number of or-
der one when R is of the order of R, the range of a loop
(R=4Bre*/?=20.8km), It is for this reason that the
factor R has been explicitly separated out in Eq. (95).

The quantity (j"!) represents the average of Gu
weighted by the internal wave spectrum H(j). We have

GL tar) ye > (FA)

= 0.730, 0. 647, 0.519, 0.435, 0.379,0.340 (99)

for j,=0,1,...,5. An approximate expression is
log (47% +1)/(nj, - 1).

For axial rays, 6=A4=0, f,(0)=1, and n=n is a con-

stant. Hence F,(R) becomes simply proportional to R;
we have
4 nv R R=
F, = ==, =e A 100
i(R) = mah 436 5 F,(R) (100)

For upward rays turning near the surface, the major
contribution to the integral defining F,(R) comes from
the ray apex (Fig. 1); here the equation of the ray is,
approximately,

z(x) 2 — (1/2R)(x- x) ,

and

224

~16°

“14° aizs

lower loop

upper loop

FIG. 3. Plots of F,(y) from a numerical integration of Eq. (113), Re F;(y) is proportional to the loop contributions towards (X2),
Re F,(0)=F} corresponds to Eq. (96) for a single upward (6 positive) or downward (6 negative) loop, The short-dashed curve

gives the apex approximation [Eq. (101)] to Fj.

A=n(x— %)/w 4 R ,

where (%, 2) is the position of the apex and ® is the ray
curvature at the apex.
ly with increasing 4, and therefore we can write, ap-
proximately,

F(R) = F(R) -4, LE ( payar-2™ =~ aor
Lee ee) T wing R jf * Rn

times the number of upward loops.

For a single complete upward (downward) loop with
range R*(R’) we denote F,(R*) simply by F?. Then for
a complete double loop, with range R*, we have

Fy =F,(R*)=Fi+F].

If the double loop has an apex near the surface, then
Fj~F, and F;~0; thus

Fy F, .
The results of a numerical evaluation of Fj and Fy
as a function of ray angle @ are shown in Fig. 3, as is

the apex approximation Fi which is seen to be an ex-
cellent approximation for 62 5°. The largest value of

The function f,(4) cuts off rapid-

)

Fj occurs at a ray angle @ near 2°, and a corresponding

225

apex depth of 750 m; deeper rays are reduced by the
smaller value of n°, rays of shallower apex are reduced
by a smaller radius of curvature R,

We shall also need the variance of X = dX/dt; this is
found by inserting w? in Eq. (66) under the integral sign.
The result is

(|X|) =wyano( (5C/CR) (77) PBRF? , (102)
where
= te n
RF, =87°°ng? | dx sec’ noe (J ) fla) (103)
0 Jin
and
Ta Be
N= BUA 2 SU rman
Fal ) (ioe Win 2 10g 4
1 1 (VAS) a es
ao + A272 og (a +A?) P24 Hi og Win
(104)

For the axial ray (where f,=1) and near surface rays
we have, respectively,

Fy = 817 (2/ng)*(R/R) logn/w,, =0.132R/R ,
F3=812(2 7)! /? (Bl QI )t/? (R)7 (/n9)8 log (/w4n) -

(105)
(106)

upper loop

lower loop (2)

FIG. 4, Plots of F,(y), proportional to the loop contributions
towards (X*), See Fig. 3.

F, is less peaked at the apex than F, and decreases
monotonically with apex depth (Fig. 4).

Let us next turn to the quantity (X°). From Eqs. (75)
and (91) we find

(X?)=— ((5C/C)%) (j*) @?BRG,(R) , (107)
where
Alpe :
Gy(R)= a RL dvsecten'h(alelAs i.) - (108)
Here we have defined
iyi 1
bal =) = Soe iBi") , 109
&(B, 5, F ds Pap ey) (109)

and we recall from Eq. (76) that
B= (1/BY (n/m! 1/qAg, «

Note that as 8-0, g(8,j,)—1 and thus G,— F, and hence
(X?)——-(1X1?),

An approximate analytic expression for g(8, j,) is

. \_Lexp(- ifj%) Ei(ip Gs +4)) — EilG-8)]
818, jg) = log (472 +1)

Thus for small 8, we have
2

j
28, 5*)=- FPHeagoay lel tees

and hence for small £,

iil ie 1 wife
G,(R) = F(R) - = a =
Wades tes.) t (j*) mj,-1 wae Ry Le

xsec’@n5f,(A)| Bl +++.

For very long ranges where many loops are involved,
we may simplify the integral in Eq. (108) as follows.
First replace 8 by the many loop long range value By;:

Bur, = (1/B)* (n/no)* [x(R - x)/qR] tan?0/5 = ayyy(x) ,

(111)
where

y(x)=x(R-x)/B’Rq . (112)

Next note that y varies rather little over one loop.
Then the integral from 0 to R may be broken up into a
sum of integrals over each of the loops. In any given
loop, say the kth one, y has very nearly the value »(x,)
=y,, where x, is the position of the midpoint of the
loop. Thus we may write, in place of Eq. (108),

+ K~
Gi so Firs) +2 Fily,) ;
R=1 R=

where A* is the number of 22%,
4 1 il XptR*/2
Tee
i(%) T wyane R

loops, and where

xp R*/2

x sec”On°f, (A )e(B, , i ,) ? (113)

with
By = 7° (n/n9)* (tan?0/5)y» «

Here x,+R*/2 are the positions of the two ends of the
loop. We note that as y,~0, Fi(},)— Fj as defined
earlier.

The variation of Fj(y) with y is shown in Fig. 3.
Large deviations from Fj begin to become apparent
when y is of order one. The maximum y that occurs
over a range R is R/4B°q; thus G, is not very different
from F, until ranges of order 4B’q.

All of the foregoing results and definitions may be
summarized in the following relation:

SO GC,

+( |x|?) 5CY F,

_([(5L 1 a
= (xX?) -(( ai) GER WiaMoGe | * ne
+(|X|?) WyNoF2

The quantities of actual interest to us are not quite
(x?) and (|X|*), but rather the mean-square phase and
intensity fluctuations. These, we recall from Eqs. (4),
(6), and (9), are related to (X*) and (|X|?) through

(7) =3((| X|?) - Re(X?)) (115)
and
(?)—(e)? =2((| X|?)+Re(X?)). (116)
Thus we find
OG = ((5C/C)) Gj) q BR(? (F,+ReG,) .
(as e (117)

For small 8, ReG,— Fy, so that

(6?) = ((8C/C)R) (77) @ BRE; ; (118)

this is simply the conventional geometrical optics ex-

pression for phase fluctuations. Intensity fluctuations,
however, depend on the difference between F, and ReG,,

4 1

alae 3 ;
ee ChE oa at dxsec’6 nf, (4)(1 - g(8, j,))

(119)
and thus vanish in the small 6 limit. Indeed, for small
B, using Eq. (110), we find

BN ante cy \ ik q ‘
een? -4n(C), our. a

2 5
A
xsec*dn°f, (4) a

(120)

and this does not coincide with any geometrical optics
expression.

It is also of interest to compute the spectra of phase
and intensity fluctuations. For this purpose, we return
to Eqs. (66) and (75), and to Eq. (91), but we do not now
carry out the integral over dw. Fora given value of w,
we integrate over a ray path, keeping in mind that there
is a complicated set of forbidden ray sections, depend-
ing on the value of w relative tow,, # , and #* (Fig. 5).
For very low frequencies the ray is too steep to permit
“stationary-phase interaction” with internal waves.

For the high frequencies w may exceed n(z) along some
portions of the ray, and internal wave solutions do not
exist. The phase and intensity spectra are given by

ere] - A (S2))(2) 48g [esc

wee wt tl?
x (—H) H(n-w)H(w- w,)
L

ea a ie: (121)

2(1 — Reg(8, jx))

For the important range w, <w<mn the entire integra-
3

tion path is permitted and the spectra vary as w~.
Vill. COMPARISON WITH NUMERICAL
EXPERIMENTS

As afirst application, and test, of the results we have
obtained we shall make a comparison with a set of
“numerical experiments.” ° These consist of numerical
solutions of the parabolic wave equation in the same
sound channel we have discussed here, and with a se-
quence of internal wave realizations from a two-dimen-
sional projection of the spectrum described in Sec. vie?
The “numerical experiments” use an acoustic fre-
quency of 100 Hz, and propagate sound up to ranges of
100 km; the remaining parameters are the “standard
ones” listed in Secs. IV-VI. In all cases the acoustic
transmitter is located on the sound axis, at a depth of
1000 m. The receiver consists of a vertical array of
hydrophones, 700 m long, centered on the ray in ques-
tion, which allows an angular resolution of 13°.

227

(Wi)max< W< a

FIG. 5. Internal wave contributions toward frequency spectra
of acoustic phase and intensity come from “permitted” sec-
tions of the ray path (heavy lines), There are no internal
wave contributions to frequencies less than the inertial fre-
quency (I) and larger than the apex bouyancy frequency (V) be-
cause no such internal waves exist. The entire ray contributes
toward the central band III between (wy)max (typically 10 «,,)
and the buoyancy frequency at the lower turning point (for deep
rays III does not exist). For lower frequencies, the upper
sections of the ray are too steep to permit “stationary-phase
interaction” (II), and high frequencies exceed the buoyancy
frequency of the deep ray section (IV).

A. Phase fluctuations

Solid lines in Fig. 6 show the results of the “numeri-
cal experiment” for 128 realizations (to which one may
assign a statistical error of perhaps +20°). The dotted
lines are the predictions of the theory outlined in Sec.
VII, and specifically of Eq. (118). Evidently the agree-
ment is satisfactory. Overall magnitudes differ be-
tween theory and experiment by about (20-30)% (except
for the —1° ray) and the general shapes coincide as well.
For the steep rays (+9° and to a lesser extent +5°) the

FIG. 6. Comparison of cal-
culated (dotted lines) and
“experimental” (solid lines)
rms phase fluctuations at
100 Hz as function of range
for six rays with inclinations
on the axis ranging between
+9°, Lines connect calcu-
lated values, with no attempt
at interpolation.

rms cycles

rms phase is nearly a step function of range, reflecting
the fact that the major contribution to the integral in
Eq. (96) comes when the rays cross an apex, and that
there is little contribution while the ray is deep. The
near axis rays (+1°), on the other hand, vary much
more smoothly with range (nearly like Ri!)

B. Intensity fluctuations

Table III shows the rms intensity fluctuations for the
same six rays, at various ranges. Since the numerical
experiment makes use of a vertical beam former rather
than a single hydrophone to select different rays, the
theoretical calculations described in Sec. VII must be
somewhat modified. There intensity fluctuations were
calculated for a fixed receiver position; here we must
calculate fluctuations for a fixed receiver angle but hav-
ing a variable vertical position. This amounts to re-
placing the quantity A in Eq. (120) by a different geo-

228

metrical factor Bjy, defined to be the second derivative
with respect to z of the optical path length from the
transmitter to a receiver located at a fixed range and
seeing a fixed vertical angle, rather than one located at
a fixed range and height. This quantity has been eval-
uated numerically, and then Eq. (120) has been used,

in order to obtain the theoretical values shown in the

table.

The quantity Be can be evaluated analytically for lin-
ear and quadratic sound channels (with circular and
sinusoidal rays) which approximate the real sound chan-
nel for deep and near-axial rays, respectively. We
find

x 4 1 sinKxcosK(R-— x)
re &

Beas Bier cosKR (122)
for deep and near-axial rays, respectively, where 27/K
is the wavelength of the sinusoidal rays, so that 7/K is
the range of one loop. We remind the reader that

a _x(R- x)
zen R ?

. _1 sinKxsink(R- x)
7K) ysinkR

A A

are the corresponding quantities for a fixed-point re-
ceiver. Thus, for near-axial rays, Bz, becomes in-
finite when the receiver is located at the turning point

of the rays. It is here that all rays are parallel; this

is the analogue of a caustic for a beam former receiver.
As we have already remarked in Sec. III, when the re-
ceiver is placed near a caustic, our approximate expres-
sions [Eq. (75)] fail; to correct it would require keep-
ing the effect of horizontal spreading in Eq. (74). We
are therefore able to compare our calculated values
with the “experimental” ones only if we avoid placing
the receiver near a (beam former type of) caustic. For
near-axis rays, where we can use Eq. (122), these caus-
tics occur at ranges R= (n+3)n/K; since 1/K= R=20 km
for near-axis rays, there are caustics of ranges of 10,
30, 50,...km. For off-axis rays, the positions of caus-
tics must be determined numerically. The entries in
Table III are made by avoiding these.

The agreement between theory and “experiment” is

TABLE III. RMS intensity fluctuations in dB, The upper
(lower) number in each entry is the theoretical (“experimental”)
value,

Range (km)
Ray angle 20 30 40 50 60 80 100
+9° 1.00 1,21 1.34
(0.73) (0.82) (1. 20)
ee 110) oe 1512) 91,89) ast
(0. 44) (0.50) (0.62) (0.64)
AE 0:79) 108) 1.3L) ot, 57 eedeeO
(0. 34) (0. 41) (0.60) (0.74) (0.95)
ae 0.66 || TOL ans 1539) 1701 2520
(0. 30) (0, 43) (0.61) (0.56) (0,46)
ug (OHS8iaem 0940 ke 1.19 1.29
(0. 41) (0, 61) (1,02) (0, 84)
ce (Os09)) JO.disiay (0129) Bae re 0.78
(0.06) (0.07) (0,31) (0, 63)

5
—CCCc—uwnaounQqQqQqqueeeeeeeeeeeee—eeeeeeSSsSsSsSsSsS oom

TABLE IV. The measured rms values of
travel time and intensity,

Frequency (nominal) 4 kHz 8 kHz
Travel time 0.384 0.374 msec
Intensity 5.2 5.7 dB

satisfactory for the off-axis rays but for near-axis rays
the theory seems to overestimate the size of the fluctua-
tions; in particular, the +1° rays are predicted to have
fluctuations that are larger by a factor of two to three
than the “experiment” shows. This discrepancy is pos-
sibly related to the fact that we use the linear approxi-
mation to the dispersion [Eq. (92)], or that the WKB ap-
proximation underlying the theory does not allow the re-
duction in vertical displacement near the boundaries,

or to the two-dimensional character of the “experi-
ment,” or, finally, to the failure of the expansion in
power of acoustic wavelength over the vertical correla-
tion length of the fluctuations at this wavelength.

IX. SINGLEPATH EXPERIMENT ON COBB
SEAMOUNT

Ewart!! has measured amplitude and phase (transit
time) fluctuations between a fixed transmitter and re-
ceiver on Cobb Seamount (46°46' N, 130°47' W). The
sound axis is shallow, 400 m, as is characteristic of
high latitude, Setting z=— 0.4 km in Eq. (84), we con-
struct a ray path through source and receiver (both at
1000 m depth) separated by 17.2 km, with a lower turn-
ing point at a depth of 1350 m, in agreement with ray
tracing based on locally measured sound profiles (Fig.
4, Ewart), Further, the measured n(z) is very close
to our experimental model [Eq. (82)]. Ewart obtained
144.5 h or records (with minor gaps) based on 8-cycle
pulses at 4166 Hz and 16-cycle pulses at 8333 Hz trans-
mitted alternately every 15.7 sec. The measured rms
variations are given in Table IV.

Ewart remarks on the strong tidal contribution to the
travel time spectra, and on the important effect on in-
tensity by sporadic multipaths associated with sound
velocity fine structure. We note that the results are
similar gt the two frequencies (as expected); the rms
phase at 4 kHz is 3. 84x10 sec x4166 Hz=1. 60 cycles.

The maximum value of 8 is 10° (7 km from turning
point), so geometric optics applies, and according to
Eq. (118)

(6?) =((6C/C)R) (77) g?BR F,(0) =251.7 rad?

for rms (6C/C))=4.9x10%, (j71)=0,435, q=1.745
x10* radkm” (for 4166 Hz), B=1km, R=20.8km, ng
=5.2x10 sec™, w,,=1.06*10% sec™ (46. 75° latitude),
and F,(0)=0.38 (from a numerical integration), Thus
rms $=2.53 cycles, compared to 1,60 measured.

Similarly the intensities are found from Eq. (120),
using Aj} = x(R - x)/R which is appropriate for a single
lower loop. The result is (c?)—()*=0.245, or

(10/log10) (0, 245)'/? =2. 15 dB,

229

TABLE V, Variation with model parameter j,.

observed ip j,=4 ea)
rms ¢ in cycles 1.6 2.5 202 2.0
5.5 2.2 2.3 3.4

rms tin dB 5 ° A

as compared to the observed rms value of 5.5 dB. (A
more accurate form for Aj} will increase the calculated
value slightly. ) Observations and computation of both
phase and intensity are roughly within a factor of 2 and
can be brought into better accord by increasing the
model parameter j, (Table V).

A more sensitive test consists of comparing com-
puted!* and observed spectra (Fig. 7). The computed
phase spectrum is high, as expected from the rms val-
ues, but in the principal band between inertial and
buoyancy frequencies the computed w~° slope is reason-
ably consistent with the observed spectral slope. The
observed phase spectrum continues smoothly beyond the
computed n cutoff. Computed intensities completely
fail to account for the observed high frequencies,
Dashen (private communication) has demonstrated that
the high-frequency phases and intensities are due to in-
terference between “sporadic multipaths.” (Ewart has
remarked on the occasional arrival of multiple pulses. )
A discussion goes beyond the scope of this paper. *®

xX. MULTIPATH EXPERIMENT MIMI

The most persistent measurements of ocean propaga-
tion are the 406-Hz transmission of MIMI"! between
Eleuthera (Bahamas) and Bermuda. The measured ¢(f)
and c(t) are completely dominated by the effects of mul-
tipath interference, and are not simply related to the
o,(t) and c,(t) along any singlepath 7 with which our pa-
per is concerned. However, it is possible to use the
measured multipath spectra to infer rms , for a typi-
cal singlepath.’® Results are given in Table VI.

For a “back-of-the-envelope” comparison (after two
years) with our results we uSe the axial approximation
(105) in Eq. (102):

($6?) =((6C/C)R) (77+) G? BR wignoF 2

= Bn? ((6C/C)ea45) GG?) @?BRw gM lOgn/wy, +
(123)

Using g=1701 radkm™ for 406 Hz, rms(6C/C))=4.9
x1o+, (j77)=0.435, B=1 km, w,,=7.3x10° sec", ng
=5.2x10°% sec?, 7=1.9x10°% sec! and Eq. (89), this
simple expression leads to excellent agreement with the
measurements (Table VI). For the surface limited ray
we use the apex approximation (106) with #=m) and a
radius of curvature R=13.7 km [Eq. (85)] to obtain

($?) = 80? (2 n)/? ((8C/CR) (57 FP BR?
X Wao LOg (129/ wn)
per double loop, leading to somewhat larger values.

Table VII summarizes a more precise calculation,
allowing for the proper “ray mix.” From Eqs. (102)
and (103)

24h 12.4h

102 103
10 10?
1 10
c 105! 1
Qa
(©)
a™~
rh To 1071
oO
~>
[S)
1075 10-2
1Ors 10-3
Urs 10-4
10-6 1055
01 al { 10 100 .01 1 1 10 100

cph

FIG. 7. Computed (smooth curves) and observed spectra of phase (left) and intensity (right) at Cobb Seamount,
results are shown for phase; at 8 kHz computed and observed values both are higher by 6 dB.

Only the 4-kHz

(Bi) = ((E),) G2) PBR are sah,

Ke
x 20 I[FHO)+F4,)] -
k

The summation 7 extends over 34 rays in the order of
decreasing inclination 9, of the axial source. For even
numbers of loops, values of +6; pair with precisely the
same Statistics, but for odd numbers of loops there is
an extra upper loop for positive 6 and an extra down-
ward loop for negative @. Each ray is weighted accord-
ing to the difference Ad, between adjoining rays, and
this emphasizes the near axial rays.

The number of loops varies from 60 (K*=30, K~ =30)
for the near-axial rays (@=+1°) to 44 (K*=22) for the
surface-limited rays. The largest y, is at the central
loop, +=3R, y=R/4B’q=0.184 and even then the differ-
ence between F,(y) and F2(0) is slight. We may then
use the geometric optics formula

Gn-(E]) Genta 3

x[K*F3(0)+K7F3(0)] .

The sumand (last column of Table VII) is fairly uni-
formly distributed among all contributing rays.

The agreement between computed and measured val-

ues of (6?) is rather too good.

TABLE VI. Measured and computed MIMI parameters,
Midstation Bermuda
Range 550 km (nominal) 1250 km
Number of paths 14 34
Number of double loops
Surface limited ray (SLR) 10 22
Near axial ray 13 30

rms ¢$, inferred from MIMI 2,8x10™ sec*

rms 6, computed
Axial ray
Apex approximation, SLR
Weighted average

2.9*10% sec"!
4.610% sec?!
3.5107 sect

4.0x10° sec?

4,4x 107 sec

6.8107 sec*
5.2107 sec?

230

TABLE VI. Calculations of ($3) for Bermuda. 4; are the
inclinations at the axial source of all possible rays to an axial
receiver at 1250-km range, consisting of K* upper loops of
range R* and R™ lower loops of range R™ (see Fig. 1). F3(0)
are the dimensionless contributions per ray loop to (4) as
read from Fig. 4, leading to the dimensionless weighted sum
(A0/ZA0)(K*F3 + K-F2).

6, Ke R K R- Ae Fy F; Sumand
12.7, 22 12.8 22 44.0 0.60 0.703 0,011 0. 362
12.3 23 Poesy 22 43.3 0,60 0,680 0,012 0, 367
11,5 23 13,2 23 41,2 0.60 0,630 0, 014 0,342
11,2 24 13.4 23 40,4 0.55 0,614 0,015 0,319
10,3 24 13,7 24 38.4 0.55 0,560 0,018 0.293
10.0 25 13.9 24 BIA 0.55 0,545 0.019 0,298

9.2 25 14,2 25 35.8 0,55 0.500 0,021 0,276
9.0 26 14,4 25 35.0 0.55 0.490 0,022 0, 281
8.1 26 14,8 26 33.3 0.60 0,448 0,028 0, 286
7.8 27 15,1 26 32,4 0.60 0.435 0,029 0,288
6.9 27 15.5 27 30.8 0,60 0,395 0,035 0. 268
6.5 28 15,9 27 29.8 0,65 0,375 0,038 0,288
5.7 28 16,4 28 28.2 0,70 0, 343 0.045 0.292
5.1 29 16.7 28 27,4 0,85 0,315 0,050 0.344
4.0 29 Wy BY 29 25.6 1,05 0,274 0,062 0,394
3.0 30 18.2 29 24.3 1.50 0.235 0.073 0,529
1.0 30 19.8 30 21.9 2,00 0,165 0,107 0.628
-1,0 30 19.8 30 21.9 1, 80 0.165 0.107 0,628
-2.6 29 18.5 30 23.8 1.50 0,219 0,078 0,501
-4.0 29 17.5 29 25.6 110) 0,274 0, 062 0.394
-4.8 28 16.8 29 26.9 0,85 0, 300 0,054 0, 326
-5.7 28 16.4 28 28,2 0.70 0,343 0,045 0. 292
-6,.2 27 16.0 28 29,2 0.65 0.365 0,040 0,274
-6.9 27 15.5 27 30.8 0.60 0,395 0.035 0, 268
-7.4 26 15.3 aul 31.6 0,60 0,418 0,031 0,270
-8.1 26 14,8 26 33.3 0.55 0,448 0,028 0,286
-8.5 25 14.7 26 33.9 0.55 0.469 0,025 0. 262
-9,2 25 14.2 25 35.8 0.55 0.500 0,021 0,276
-9.6 24 14.0 25 36.6 0.55 0,512 0.020 0,271

-10.3 24 13.7 24 38.4 0.55 0,560 0,018 0,293

-10,6 23 13.6 24 39.0 0.55 0.585 0,016 0, 293

-11.5 23 13,2 23 41,2 0.60 0.630 0,014 0,342

-11.8 22 USL 23 41.8 0,60 0,652 0,013 0,338

-12.7 22 12,8 22 44.0 0.60 0,703 0,011 0, 362

XI. CONCLUDING REMARKS

We end up, after lengthy derivations, with quite sim-
ple and transparent formulae for the acoustical fluctua-
tions. The formulae make explicit the dependence of the
various oceanographic and acoustic parameters. The
need is to apply these results to a variety of experimen-

tal situations.

For Project MIMI the measured acoustical fluctua-
tions are dominated by the statistics of multipath inter-
ference. The observations yield but one parameter
which is sensitive to the ocean model: (¢? ). Values at
midstation and Bermuda are close to those computed for
an internal wave model based entirely on oceanographi-
cal observations. (There are no free factors in this
comparison, ) The agreement could be made even closer
by a reasonable adjustment of internal wave param-
eters. We conclude that internal waves play an impor-
tant and probably dominant role in producing the acous-
tic fluctuations.

The MIMI transmissions are characterized by many
deterministic multipaths as determined by the gross
profile C(z); the statistical results are not affected by
the additional sporadic multipaths resulting from a fine-
structure 6C. In contrast, the Cobb Seamount experi-
ment has a single deterministic path, but because of the

high acoustic frequency, sporadic multipaths play an
important role in producing high-frequency fluctuations
of intensity and phase. Dashen (private communication)
has shown that an extension of the present analysis,
based on the same ocean model, can account quantita-
tively for the high frequencies in terms of sporadic
multipathing, but this goes beyond the scope of our pa-
per. The mean-square quantities, in contrast, are
dominated by low frequencies and can be estimated
from singlepath theory. We find measured and computed
rms fluctuations to be within a factor of 2.

ACKNOWLEDGMENTS

This work has been strongly dependent on the closely
related efforts of R. Dashen and S, Flatté. The theo-
retical framework was Set in an earlier report written
by C. Callan and F. Zachariasen. We wish to express
our gratitude to Callan, Dashen, and Flatté.

*This work was performed during the 1974 and 1975 JASON
Summer Studies under the auspices of Stanford Research In-
stitute, supported by the Advanced Research Projects Agency,

‘Using €=10~4 cm’ sec™ for the dissipation per unit mass, and
n=107 sec! for the buoyancy frequency.

2c, J. R. Garrett and W. H. Munk, “Space-Time Scales of
Internal Waves,” Geophys. Fl. Dynam, 2, 225-264 (1972),
C. J. R. Garrett and W. H. Munk, “Space-Time Scales of
Internal Waves: A Progress Report,” J. Geophys. Res, 80,
291-297 (1975); J. L. Cairns and G. O, Williams, “Internal
Wave Measurements from a Midwater Float II,” J, Geophys,
Res. (1976)(in press). The model initials suggest some
planned obsolescence and have allowed the authors to bring
out new models from time to time,

3C, G. Callan and F. Zachariasen, Stanford Research Institute
Technical Report No. JSR-73-10, April 1974 (unpublished).

‘See L. Chernov, Wave Propagation in a Random Medium
(McGraw-Hill, New York, 1960); and V. I. Tatarski, The
Effects of the Turbulent Atmosphere on Wave Propagation
(unpublished),

5We do not have to be very precise about how we define this
angle.

®W, Munk, ‘“‘Sound Channel in an Exponentially Stratified Ocean,
With application to SOFAR,” J. Acoust, Soc. Am. 55, 220—
226 (1974), Fig. 1 (based on Pingree and Morrison). (In that
paper, z is positive downward.) See also Fig. 1 in GM72,
Ref. 2,

7J, Roberts, University of Alaska IMS Report No, R73-4 (1973)
(unpublished).

83. Cairns and G. Williams, “Internal Wave Measurements
from a Midwater Float II,” J. Geophys. Res. (in press)
(1976).

3s. M. Flatté and F. D. Tappert, “Calculation of the Effect

. of Internal Waves on Oceanic Sound Transmission, ” J. Acoust.
Soc. Am. 58, 1151-1159 (1975).

!0The numerical experiment uses the exact wave functions for
the exponential model ocean, weighted according to Eq. (90),
whereas our analytical model is based on the corresponding
WKB approximation,

1p. &. Ewart (unpublished).

'2The computed phase spectrum [Eq. (119)] is in units of rad?/
rad/s); multiply by (27)*/(3600/27) to get cycles*/cph, For
the intensity spectrum [Eq. (120)] multiply by (10/log10)*/
(3600/27) to get dB’/cph,

13Roger Dashen has pointed out that the shallow sound axis
(2=—0.4 km as compared to the canonical 7= —1 km) can be
expected to produce smaller 6C/C fluctuations, given a
canonical internal wave field. The reduction is proportional

231

to the potential velocity gradient 8,C, [Eq. (87)], and hence For a recent paper see J, G. Clark and M. Kronengold,

to 77? ~e®2/8 (qs. (79), (81)]. The reduction in rms 6C/C “Long-period Fluctuations of CW Signals in Deep and Shallow

is by a factor 3.3. This would more nearly align observed Water,” J. Acoust. Soc, Am, 56, 1071-1083 (1974).

and computed phase spectra, but at the expense of an even 15%. Dyson, W. Munk, and B. Zetler, “An Interpretation in

larger discrepancy in the intensities. Terms of Internal Waves and Tides of Multipath Scintillations
‘4 Por Miami-Michigan project, starting with J. C. Steinberg Eleuthera to Bermuda,” J. Acoust. Soc. Am, (1976) (in

and T, G, Birdsall, ‘Underwater Sound Propagation in the press).

Straits of Florida,” J. Acoust. Soc. Am, 89, 301—315 (1966).

232

INTERPRETATION OF MULTIPATH SCINTILLATIONS ELEUTHERA
TO BERMUDA IN TERMS OF INTERNAL WAVES AND TIDES

Freeman Dyson

Walter Munk and Bernard Zetler

Reprinted from the Journal of the Acoustical Society of America
Vol. 59, No. 5, May 1976.

233

Interpretation of multipath scintillations Eleuthera to
Bermuda in terms of internal waves and tides*

Freeman Dyson

The Institute for Advanced Study, School of Natural Sciences, Princeton, New Jersey 08540

Walter Munk and Bernard Zetler

Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, California 92093

(Received 10 December 1975)

Rate-of-phase and intensity spectra due to time-varying multipath interference depend essentially on a

single parameter v* which can be interpreted as the mean-square rate-of-phase for any typical single path.
MIMI 406-Hz phase and intensities are consistent with v_' = 270 and 357 sec for Eleuthera to Bermuda and
Eleuthera to midstation transmissions, respectively, compared to 192 and 286 sec from a ray-geometric
calculation using an internal wave model based on oceanographic observations. Internal tides play a

significant but not dominant role.

Subject Classification: [43] 30.20,[43] 30.35.

INTRODUCTION

The purpose of this paper is to compare some statis-
tical properties of random vector fields with measured
cw transmissions of MIMI! between Eleuthera (Baha-
mas) and Bermuda. (Among the previous analyses we
refer particularly to the work of Clark; Dyer; DeFer-
rari; and Jacobsen.*) The observational material, gen-
erously made available to us by John G. Clark, con-
sists of intensity J(t) (decibels, arbitrary reference)
and phase &() (in cycles), as presented by the top two
curves of Fig. 1, a selected portion is shown point by
point in Fig. 2. The observed acoustic pressure fluc-
tuations (frequency o=406 Hz) relative to some (refer-
ence) scale fg can be written

P(t)/po = x(t) cosot +y(¢) sinot,

where x, y are slowly varying (compared to o) ampli-
tudes.*° The original measurements consist of the 5-
min averages‘ of amplitudes

6t
x)= f at= RU) cospts) ,
0
(1)
Y(t)=R(¢) sing(t) ,
which are related to the plotted time series (as fur-
nished to us) according to
I= 20 logR ,
6=¢/2r.

(2)

We have then reconverted to

X=10!/*° cos276,
(3)

Y=10!/° sindro
For statistical theory, it is convenient to refer to

1n10

= aie
fee aro

Ties (4)

A suitable intensity reference is

tg =1n(R2) or Ip=10 log(R?) .

|. MEASURED VARIANCE IN MULTIPATH PHASE
AND INTENSITY

Only fractional cycles are measured, and there is an
ambiguity concerning the integer number of cycles.
Normally this can be resolved by the continuity of the
time series. Phase difference over the sampling in-
terval 6¢=5”" has an rms value of 56=0.24 cycles, at
Bermuda, and the (Gaussian) probability for |56| to
exceed 3 cycle is 4% (8% were observed). A restric-
tion to |5@| = 5 cycle (which can be attained by adding
and subtracting integer cycles) is not realistic. We
have edited the observations to remove phase “kinks, ”
replacing the reported value @ by 6+1, 6+2,... cy-
cles when required to make the adjusted phase differ-
ence 66,=,,, —®, subject to the restriction

|66, —4(56,,,+5,_,)| = 4 cycles.
1

This is essentially placing an upper limit on second dif-
ferences in phase; 5% of the Bermuda observations and
1.5% for the midstation were adjusted accordingly.
Figure 3 shows a sample of 5@ before and after adjust -
ment, and Fig. 4 the reconstituted 6=)5@, Midstation
phases are not severely altered by phase adjustments.
At Bermuda the low (week-to-week) frequencies bear
no resemblance to the midstation trend and are con-
Siderably altered by the phase adjustment; however,

the high-passed records (tidal frequencies and higher)
are not significantly altered. We conclude that sam-
pling was adequate for midstation phases and high-fre-
quency Bermuda phases, but that sampling was not ade-
quate to obtain low-frequency trends at Bermuda. Ad-
justed mean-square phases and phase differences are
given in Table I,

Multipath intensities are characterized by occasional
deep fades (Figs. 1 and 2). They are censored for a
subsequent analysis of fade statistics, by replacing the
recorded values of J by J) — F whenever I[<I)—F, but
otherwise leave J unchanged; censored X and Y are sub-
sequently computed according to Eq. (3). Accordingly
the three columns in Table I refer to the removal of
fades exceeding F=~, 20, 10 dB (the first column then

235

140

eo eter ye

i eg W

Lad Bs arog

[S)

of NAVA ae AVAWAVAYIVAAV CASI Ava caT NYAS VICAAVAUAGYA data icae st Cae AW ON CA

FIG. 1. Plots of measured inten-
sity J, phase 4, and high-passed
phase, and of the components X
and Y (in arbitrary pressure units)
of acoustic pressure at Bermuda,
22 Sept.—-17 Oct. 1973.

107— Y,
ok
| | tM} ! Wel
tov !
U 1 1 J rt J t 4 i Jt 1 4 oe J
10) 2 4 6 8 10 12 14 16 18 20 22 24 26
Time (days)
refers to the uncensored record). A typical signal-to- multipath statistics.
noise ratio is 27 dB [Ted Birdsall (personal communi-
cation)], and a removal of deeper fades (say F= 30 dB) The crucial importance of sampling needs to be
would be associated with noise statistics rather than stressed. From a numerical experiment (Sec. VII) we
140

cycles

Time (hrs)
FIG. 2. A 14-h sample of intensity J and phase @ drawn on an enlarged scale,

236

CYCLES
=

op)
: 140 A il, ty vl
oO
> Vw
=
SAMPLE
170 1800
[+4 + ph tt tt
ie) 5: 10 15 20 25
HOURS
FIG. 3. Bermuda phase differences 6@ before (top) and after adjustment (bottom),

learn that sampling at 3-min intervals would have
avoided all ambiguities. As it is, the 5-min averaging
suppresses the high frequencies, and the phase ambig-
uities dominate the low frequencies.

Il. THE STATISTICAL MODEL

We shall compare the observed data with a crude

statistical model of a multipath acoustic signal. In the
model, the components of the signal are
n
Dee Xin,
Fi (5)

n
Y= Sey, ,

isl

resulting from the superposition of n single-path com-
ponents

X,=R,cosd; ,
Y,;=R; Sind;

(6)

The amplitudes R; and phases ¢, of the single-path com-
ponents are independent random variables. We use the
notation ( ) to indicate a statistical average over all sin-
gle path signals simultaneously, while ( ),; denotes an
average over the A; and ¢; in one Single path. We

make three further assumptions concerning the single-
path signals:

(1) Fluctuations in phase are more important than
fluctuations in amplitude, or in symbols (with dot ac-
cent designating d/dt)

(R32); <«K (RD; (63); (7)

(2) The time scale (¢,) for a phase ¢; to change by
1 rad is on the average short compared with the time
scale ($,/¢,) for the phase variation to change direc-
tion, or in Symbols

($3); = ($4; (8)
(3) Each phase velocity o; is a Gaussian random vari-
able, so that
($3)1= 36)? - (9)

In Appendix A it is verified that property (2) holds for
the single-path phases predicted by the ocean model
which we describe in Sec. IV. Properties (1) and (3)
also hold in the same model. We suppose that the fre-

—— i)
F
Me

BERMUDA

MID-STATION 4

VWWAn om
wv vy eA r |

raw

heavy A
: TVA

vv . =|

Ww A, adjusted “Wyn vs

Bi Sy Ma
cycles WNW)

a8 v\ Wy,

BERMUDA HIGHPASSED wn

\ A ah Aa low A =
trv VA Ww Ww Any WNW \v NAN vy vain) Ws w\
adjusted 7

yo yew wv V\ \n Mw AN Vimy WA Vln AY Vw
a oe ee ae 4 rite EE 4 n

te) 5 10 (5) 20 25

Days
FIG. 4. Bermuda and midstation phases before and after ad-
justment.

2317

TABLE I. Adjusted phase and intensity statistics at 406 Hz for data interval 6t=5.0394 min.
The columns marked ~, 20, 10 correspond to the suppression of fades exceeding ~ (no
change), 20, 10 dB, respectively. X, Y, R=(x?+Y°)!/* in arbitrary pressureunits; to cor-
rect to absolute level (dB/uB), subtract 169.0 dB for midstation and 173.0 dB for Bermuda.

Midstation Bermuda

1250

Range from Eleuthera (km) 550 (nominal)

Record length 8255 terms= 29.0 days 7366 terms= 25.8 days

(@2), ((64)2) (cycles?) 548, 0.029 110, 0.060
F (dB) cc 20 10 co 20 10
Number of terms replaced 0 114 1076 0 if) 746
(x?) («x 1078) 8.73 8.73 8.79 0.668 0.667 0.671
cy?) (x 1074) 8.70 8.70 8.76 0.655 0.655 0.659
((6X)2) (« 10748) 6.08 6.08 6.14 0.797 0.797 0.806
((6¥)?) &« 107!) 5.92 5.92 5.98 0.790 0.791 0.799
(R*) (x 1074) 17.43 17.43 17.55 1.323 1.322 1.330
ly (dB) 142 see 131. 22 : tee
(I) (dB) 139.22 139.27 139.82 128.61 128.65 129.06
(P)-(1)* (dB?) 38.79 35.87 23.37 31.84 30.18 22.52
(6?) (dB?) 24.10 21.77 13.09 48.17 45.31 30.34
(| 6l-66|) (dBcycles) 0.55 0.52 0.38 1.15 113 0.92
quency v defined by Feel) = (21) */? (u2/v) exp(= $w?/v?) ; (16)
Ze neNe
v= (9: (10) and similarly for Fy(w). The advantage of the Car-
is the same for every Single-path signal. The root- tesian spectra over the more traditional polar coordi-

mean-square multipath signal amplitude py is defined by
pr=D_ (RD, .
1

It turns out that all important statistical properties of the
multipath signals are determined by the parameter v.

(11)

Ill. CARTESIAN STATISTICS

The statistical model of Sec. Il predicts a covariance
function for the multipath signal X which is a sum of
contributions from the single-path components, namely,

(X(t) X(¢+7)) = > (R?), (cos@ ,(t) cosd (t +7); (12)
i
Now we assume that for each single-path the root-

mean-square phase fluctuations are of the order of one
cycle or larger, that is to say

(Lo, (é)]?), > (2)?

This is not really an additional assumption but is al-
ready implied by (8). [The ocean model (Sec. IV) gives
rms $, =87 for Bermuda.] We are also assuming that
the single-path phase differences [¢ ;(¢) —@,(¢+7)] are
Gaussian random variables, with a variance

(Lo s(t) — o Mt +7)P); Sere, (14)

according to Eq. (10). Putting together Eqs. (12), (11),
and (14), we find

(X(t) X(t +7)) =4 pw? exp(- 4v?7?)

(13)

(15)

The Fourier transform of this quantity is the spectrum
of X(t), namely,

nate representation involving intensity and phase (to be
discussed later) is that the Cartesian multipath and
singlepath spectra are simply related. It is disappoint-
ing that the inequality (13) applies, so that the Carte-
sian statistics (single path or multipath) provide only
such limited information about the ocean medium,
namely, the two parameters yp andy. (This limitation
would not apply at short ranges or low frequencies. )

Figure 5 shows the Cartesian spectra with plots of
Eq. (16) drawn for indicated values of vy, An alterna-
tive method for estimating the value of v is to use the
formula

ry (x?) +(¥?)

ass) oe
The values so inferred are summarized in the first two
lines of Table I.

IV. INTERNAL WAVES

These values can be compared with those derived
from a theory of sound propagation through a fluctuat-
ing stratified ocean, Starting with a spectrum of inter-
nal waves® empirically derived from various oceano~
graphic measurements, Munk and Zachariasen® obtain

($9; = 87 a ((6 C/O)zis) q?BRw in/Zo In(taxis /w tn) Gi) (18)

for a ray along the sound axis. Here m, Mais are val-
ues of the buoyancy frequency at the surface and sound
axis, and 6C/C is the sound velocity perturbation due
to internal waves, q is acoustic wavenumber, B the
scale depth of stratification (dn/dz=—z/B), R is range,
wi, is inertial frequency, andj is the internal wave

238

cycles per day

sal 10 100
10°
MID-STATION
Nf
x rae SASS
wan
VA
y \h
10"
42k
=
jo}
ne} v
o
a
BERMUDA
ss x
2 — ¥
za Y Sea :
wo LV,
2 iy :
=f VA \
10" NS HF
art
ary
1 ae \
tor, . -
10 10 10
Hz
FIG. 5. Spectra of the Cartesian pressure components X, Y

(in arbitrary units) per bandwidth 2.58=10"° Hz. The com-
puted curves are drawn for indicated values of v~! in seconds.

mode number. An internal-wave-weighted average of
7? equals (j*)=0.44, 0.34 for a mode scale number

jy =3, 5, respectively. The values so computed are in
very close agreement to those inferred from the acous-
tic observations (Table II). A detailed calculation al-
lowing for the proper ray mix leads to somewhat larg-
er values.

The important feature is that there are no free pa-
rameters in this comparison between observed and
computed values. We conclude that internal waves con-
sistent with oceanographic observations can account for
the measured acoustic fluctuations. Considering the
idealization of the ocean model the agreement is rather
too close. In particular, the assumed exponential strat-
ification and resulting canonical sound channel fail to
allow for the important intrusion of Mediterranean wa-
ter. For further detail we refer to the original paper.

V. PHASE AND INTENSITY STATISTICS

The observed data (for example, Figs. 1 and 2) are
customarily plotted in terms of intensity and phase.
We are therefore interested in calculating the statisti-
cal behavior of « and ¢ predicted by our model, these
quantities being related to the Cartesian amplitudes by
Eqs. (6)—(8). The statistical behavior of « and ¢ is
dominated by the effects of “fade outs,” which are brief
periods during which both X and Y are small and ¢ is
rapidly changing. It is convenient to define a fade-out
precisely as a time interval in which

R<eu , (19)

239

where € is an arbitrarily chosen threshold fraction, and
p. is the root-mean-square value of R according to Eq.
(11). (The multipath intensity drop F= 20 log;)e dB,
so €=0.1 corresponds to a 20-dB fade-out.) If the num-
ber n of single-path signals is large enough, we expect
to find the statistical behavior of the multipath fade-
outs to be independent of the details of the single-path
components. We conjecture that “large enough” means
only that (1) n=3, and (2) no one singlepath component
dominates the others. The conditions appear to be am-
ply fulfilled®: »=14, 34 for midstation and Bermuda,
respectively, and the relative contributions among these
paths varies by less than a factor of two.

We assume that » is “large enough” so that the multi-
path components X and Y and their rates-of-change Ne
and Y are independent Gaussian random variables. We
then have two numerical predictions for the behavior
of cand @. The statistical variance of : is

(0?) — (0)? = 97/6 , (20)

and so rms J=767/? 10/In10=5.57 dB as compared to
the measured values 6, 2 and 5.6 dB for midstation and
Bermuda, respectively. The correlation between the
rates of change of ¢ and ¢ is
(cll) 2

Cayigayyre °° 83» i
as compared to 0.66 and 0.68 for midstation and Ber-
muda. The relations (20) and (21) are independent of
the details of the fade-outs, but to obtain further infor-
mation about the behavior of ¢ and ¢ we must examine
the fade-outs more closely. The model predicts the
following statistical properties of fade-outs.

(1) The fraction of time occupied by fade-outs is

ple)=e? . (22)
(2) The average duration of a fade-out is
tates , (23)

(3) The average interval between fade-outs is
T=[7/ple)|=20° ev) . (24)

To form an easily visualizable picture of the fade-out
process, we suppose that the signal components (X, Y)
drift past zero at uniform speed during the fade-out in-
terval. For this uniform-drift picture to be approxi-
mately valid, we require that the change in the speed

TABLE II. Comparison between mea-
sured and computed values of v=rms
oy.
v (sec™?)
Midstation Bermuda

Acoustic measurements (MIMI)

Fig. 5 2.8x10° 4,0x1073
Eq. (17) DEBnel Ogo mead On
Theory based on internal wave model°

Eq. (18) for jx=3 2.9x107 4.4x1073
Ray mix for jx=3 3.510 5.2107

TABLE III. Computed fade-out statistics,
Midstation, v!=5.9 min Bermuda, v!=4,2 min
F=0, 20, 10 dB F=~%, 20, 10 dB
Fractional time (Eq. 22 0 LO 10m 0 11072) don!
Duration (Eq. 23) 0 1.6 5.3 min 0 a 3.7 min
Interval (Eq. 24) % 165 53 min oo 111 37 min

x during the interval be less than X itself. In terms of
statistical averages, we require

TRE) (Xe) F (25)
Now our assumptions (7), (8), (9) imply

(P)=S ns Oe)= 2 ny? (26)
and therefore the condition (25) becomes

gme<1 (27)

The condition is barely satisfied with €=0.1, and so
we assume in the following discussion that €=0.1
(F= 20 GB).

The computed fade-out statistics (Table II) for 10-
dB fades do not satisfy this condition. Further, the 5-
min averages in the observations will suppress most
of the 20 dB and a good fraction of the 10-dB fade-outs.
Thus, there is little left for a quantitative comparison.
The computed durations are consistent with the observa-
tion that for midstation 92% (61%) and for Bermuda 100%
(90%) of the 20- (10-) dB fade-outs consist of single
terms, that is, the duration is less than 5 min. We ex-
pect to miss most of the 20-dB fade-outs, and many of
the 10-dB fade-outs, particularly at Bermuda. In fact,
97 (567) were observed at midstation compared to 275
(860) computed, and 34 (375) at Bermuda compared to
371 (1125) computed. All one can say is that the re-
sults do not contradict the computations, but for ade-
quate studies one will need to sample at least once per
minute.

Vi. RANDOM WALK AND SPECTRA

We picture the movement of the multipath signal
(X, Y) as a two-dimensional zig-zag random walk,
shown schematically in Fig. 6. The track is composed
of discrete straight segments of mean duration ¢, (to be
estimated). We assume that the motion in each segment
is uniform and that the tracks in different segments are
uncorrelated. .Then the behavior of the multipath phase
¢ is defined if we assign a probability distribution
Q(6)d@ for finding a phase change A¢ in the range [@,
6+d6] in a given segment of track. Values of Ag close
to +7 are associated with fade-outs.

The zig-zag walk model is not intended as a quantita-
tive representation of reality but only as a guide to the
analysis of observations. In particular, it does not
make sense to try to compute the distribution function
Q(6@) exactly. We have two pieces of information about

Q(6).

(1) The probability of a fade-out in any one segment
of the track is

240

4en/? O(n) =(t,/T) » (28)
where T is the interval between fade-outs given by Eq.
(24). We thus obtain the estimate

Q(7)= (20?) vt, .

(2) The average phase change per segment is related
to the mean value of |@| which we obtain from Eqs. (11)
and (26):

(29)

(ao)= { | ¢| Q(0)de=(|b|)t,=vt, (30)
We assume for Q(6) the simple form
Q(6)=(27)*(1+b cosé@) , (31)

and use the two conditions (29) and (30) to determine
the two parameters ¢, and). The result is

b=72(20? — 4) 74=0.63 ,

(32)
vt,=m(1—b)=1.17
The mean-square phase change per Segment is
(ag) = f 6° Q(6)d6= 51? ~ 2b= 2. 04 ; (33)

This means that the root-mean-square phase change
per segment of track is 1.43 rad or 82°. Over a time
t long compared with ¢, the mean-square phase wander
is

(oe (t) - (0) ) =avt,

_ (dg?) _ 2a(7? -5)
a Da, 3r* -12

(34)
=1.74 .

The model is of course very crude; from a numerical
experiment (Sec. VII) we find

(((t) — (0) }?) =2. 78 vt .

Over a month’s duration the expected random phase walk

FIG. 6. Random-walk model of multipath signal in the (X, Y)
plane. A fade-out occurs when the track crosses the small
circle of radius ey,

cycles per day

1 10 100

BERMUDA

ne)

fe)

a
Ny ———— SN,

a MID-STATION

12)

MID-STATION

ne)

= =
a ‘BERMUDA BERMUDA -

Qa
“

ne)

10° 10° 10°
Hz

FIG. 7. Spectra of phase difference and intensity (bandwidth
0.915 cpd). The computed curves are drawn for indicated val-

ues of v=! in seconds. The area under the intensity spectra
(e.g., the mean-square fluctuations) is independent of v.

is by (2.78 vt)'/?/27 = 20-30 cycles (for comparison, see
Fig. 4).

We next obtain from the random walk model pre-
dicted spectra for the quantities : and @. The spectrum
of the Cartesian components X, Y was already given by
(13) in Sec. ID.

The high-frequency spectra of . and ¢ are dominated
by the fade-outs. Each fade-out is approximated by a
segment of track in which the Cartesian components
(X, Y) move linearly, so that

c(t)=In[ V(t =¢,,)? +R?) ,

p(t)=arctan[V(t—-2,,)/R]+const ,
where V, R and?,, are random variables. Taking
Fourier transforms of Eqs. (35) and (36) and averaging
over the variables V, R, ¢,,, we find the spectra

F,(w) =4F, (w) =3v?w (37)

valid at high frequencies when w>v . In performing
these averages we used the probability distribution of

(35)
(36)

241

fade-outs given by Eqs. (22) and (24).

The spectrum at low frequencies will be dominated
by the phase-wandering described by Eq. (34). The
Fourier transform of Eq. (34) gives

Fy(w) = (av/t)w™ , (38)
for w<v . Inthe case of «, we expect F (w) to be finite
at low frequencies since :(¢) does not wander but re-

mains bounded as /~~, We know the total variance of
«from Eq. (20), so that

fo Flo) da =(n?/6) (39)
0
Spectra consistent with Eqs. (37)-(39) are
15,2927, 02, 2 27)-1/2
Fy(w) =20?w?[w? + cp?]-1/2 , (40)

F.(w) = 2v7[w? +120 y?]3/2

where c=37a‘=0.90. Both spectra show the expected
transitionfrom low- to high-frequency behavior at w= v.

We cannot expect this crude model to give exact quan-
titative information about the spectra. Accordingly we
modify Eqs. (38) and (37) to the form

F,lw)=aXavTtw? ,

Fy(o)=4F (wo) =pxov7w™ ,

(41)

for low and high frequencies, respectively. A numeri-
cal experiment (Sec. VII) can be fitted to

a= i 6, B = Zs 0 ’
which gives

F,(w) =v? (wu? 41.27 v2)-1/2 |

F (wy) = 4v?(w? + 2.43 v?)73/2 |
Figure 7 shows the comparison between the computed
spectra (42) and the observed spectra.” The overall
agreement is not good. The high phase values at the
lowest frequency band (over and above random walk)
could be the result of coherent modulation by large-
scale ocean features; some of it might be due to tides
(Sec. VII). The predicted w! and w~ rolloffs for rate-
of-phase and intensity spectra are borne out at midsta-

tion, The high-frequency Bermuda intensities are
aliased from undersampling.

(42)

Computed mean-square variations are

phe ror uey

@)e [~ F (w) dw = 2v? [sinha - a(1+@)1/7] , (43)
0

a= (1/23 )w'/v,
and these become logarithmically infinite as w’—+.
The upper limit is set by the integration time 6¢, and
crudely w’=27/5t. Results are given in Table IV.

TABLE IV. Root-mean-square phases and intensities (w’

=0.0208 sec"),

oO
Mid station, v!=357sec Bermuda, v~!=250seo

rms 66 rms 6/

Computed (Eq. 43) 0.19 cycles 7.0dB 0.26 cycles 8.9dB
Observed (Table I) 0.17 4.9 0.25 6.9
S06—\=e—NuNiNVNVNVQVQ0"”"0?*awx0T€=0—sa00S oeoa=oaoO=$q$q$qaDmom9SS ee”:

rms 56 rms 6/

, Fxtw, Fy (w
I — Seat Nt
eq 16 yt ee VANE E
ir 10 a>
2
fa (w) \ Fa
\ ; 2
: a
— A y
= LV SAV AVA
‘ c
3
E 10 io?
oS Be
E 3
a =
iS te
=
3
as
uw 10° 10
! ae
&
BS
\,
*
10 10° 1
radians per minute (rpm)
FIG. 8. Computer simulation of multipath statistics. The

curves labeled Fy (w) give the average spectra of the random
single-path input functions dy (t); solid: a bandlimited (2—24
cpd) uniform spectrum, and dashed: a w* spectrum above 2
epd. For both cases (dj) =1/(5 min)*. The resulting spectra
of the multipath Cartesian components X(¢) and Y(t) (solid for
w, dashed for w~*) are in good accord with the predicted
Gaussian behavior [Eq. (16)]. At high frequencies the com-
puted spectra are too wiggly to be plotted, they fall within the
limits of the shaded band.

Vil. NUMERICAL EXPERIMENT

Figures 8 and 9 show the results of numerical ex-
periments. The singlepath series 5¢,(t) were gener-
ated from random numbers for two cases: (1) a band-
limited (2-24 cpd) white spectrum and (2) an w~® spec-
trum for w >2 cpd (computed by accumulating random
5°o,). The singlepath phase series are formed by $,(¢)
=)'.95,(¢), and the multipath according to

X(t) = ah coso¢;=Rcos¢ ,
ial

and similarly for Y(¢), with R; arbitrarily set to 0.1.
Spectra were computed for X, Y, ¢,2. This computa-
tion was repeated ten times (using, of course, different
random noise series), and an average of the spectra so
obtained has been plotted. The results are essentially
the same for the w°’ and w~ spectra of }; (which bracket
the theoretical w+ spectrum®), as expected.

For both cases we have taken v?= (6%) =1/(5 min)’,

242

representative of MIMI. The input series consist of
2880 terms each, interpreted as a one-day record at
5-min intervals. This sampling rate 6/ was chosen by
trial and error to avoid ambiguities in multipath phase
during occasional fade-outs. It would thus appear that
vbt= yy would give adequate sampling for a field exper-
iment.

Figure 8 shows the average of the 100 input spectra,
and the associated Cartesian spectra according to Eq.
(16); these provide a check on the numerical experi-
ment. The spectra F.(w) and F3(w)=w*F,(w) in Fig.
9 have been fitted by Eqs. (42).

VII. TIDES

The tidal contribution to the acoustic fluctuations has
been emphasized in the literature, ® perhaps because of
a superficial resemblance of the phase fluctuation ¢(t)
to tidal records (Fig. 4). Our conclusion is that tides
play a significant but not dominant role. We shall dis-
cuss three hypotheses: a coherent modulation of the
acoustic transmission by surface tides; a coherent
modulation*by internal tides at the terminals; an inco-
herent modulation by internal tides along the entire
transmission path. Unfortunately, the evidence does
not lead to a clear-cut decision.

For orientation we have put together an order-of-
magnitude summary (Table V) of amplitudes of tides
and internal waves (a rash extrapolation of recent com-

eq. 42
E
iS
aS
1
hat
ee
i eq. 42
€ 10'
=
10° —_—_l. Ll
10° 10" !
radians per minute (rpm)

FIG. 9. Spectra of multipath intensity and rate of phase from
the numerical experiment, corresponding to Fy ,(w) ~ w? (solid)
and w~* (dashed), respectively.

1
1
=
(=o
te
a
a+
I
sk

FIG. 10. Profiles of vertical displacement ¢ (left) and hori-
zontal velocity u (right) for surface (mode 0) and internal
(1,2) tides. Scale is arbitrary.

pilations’), For the internal continuum estimates are
based on a recent version of the GM75 model.°® All
phases are considered random, and so the totals are
summations in squared amplitudes. Actual values de-
pend on the local temperature and salinity prpfiles, and
vary considerably from place to place. Mode numbers
refer tothe number of zero crossings of the horizontal
current u(z) (Fig. 10). Here we distinguish between sur-
face tides (mode 0) with a uniform current from top to
bottom, and internal waves and tides (modes 1, 2,...).
Surface tides have wave lengths of 3000 km, internal
tides 100, 60 km for modes 1, 2. Surface tides are
known to be sharply peaked at M, frequency; internal
tides are intermittent and broadened.

The sound velocity is perturbed by both vertical dis-
placement and horizontal currents:

5C/Cx=(10, 1, 0.01)x10%

for ¢=1m, at depths of 0.1, 1, 4km ,

5C/Cx0.7xX10°%, for w=1 cm/sec, at all depths

For surface tides it would appear that for a typical ray
path the w effect dominates, but for internal waves and
tides the ¢ effect clearly dominates except at abyssal
depths. Further, most of the w energy (but only a frac-
tion of the ¢ -energy) is at inertial frequencies, yet we
will show that there is no discernible inertial peak in
the acoustical spectra.

To study tidal and inertial effects we need to analyze
the acoustic records at high resolution. Accordingly
the records were divided into the initial and final one-
half months (somewhat overlapping for Bermuda), and
the spectra computed for each harmonic. In this way
the spectra are computed at precisely the frequencies
of major tidal constituents. The two spectra are com-
bined for obtaining the average power in each band,
The statistical reliability is manifestly poor; there are
only two degrees of freedom in the fortnightly analysis,
and somewhat less than four degrees in the combined
analysis. Phase spectra (Table V) show a significant
semidiurnal tidal peak, Cartesian spectra (Table VII)
do not. (spectra likewise have no tidal peak.) We
estimate 2.5 square cycles (subtracting background) in
the semidiurnal ® peak at Bermuda, as compared to a
total variance of 35 square cycles (excluding subinertial
drift), For the important 566 spectrum, tides account
for 4x10 square cycles out of a total of 60x 10°
square cycles.

The simplest interpretation is that the current asso-
ciated with surface tides (wavelength 3000 km) co-
herently modulates phase along all paths. The travel
time R/C is modified by a fraction u/C, and

46 = 270(R/C) (u/C)= 2.3 cycles,
for 270= 406 Hz, R=1250 km, u=1 cm/sec, C=1.5 km/
sec. Asa model of coherent phase modulation, set

TABLE V. Representative magnitudes in the Northwest Atlantic for the vertical displacement ¢ and horizontal
velocity u of tides and internal waves at thermocline, sound channel and abyssal depths (h=0.1, 1, 4 km). See

text.

Designation (mode) surface (0) internal (1)

internal (2) internal

I
€(m) u(cm/sec)

h(km) é(m) u(em/sec) €(m) —u(em/sec) t(m) —u(em/sec)
Diurnal tides
0.1 0.1 0.2
1 0.08 0.2 20%—30% of semidiurnals
4 0.02 0.2
Semidiurnal
0.1 0.5 1 1.4 0.9 1250.8 3 2
al 0.4 1 2.3 0.3 2 0.2 5 10,
4 (el 1 4.6 0.2 4 0.2 10 5
Inertial cusp (wy <w< 2u,)
0.1 1.8 2.8 1.6 2,4 4 6
al small elace lat} Zo LG ve 4
4 15 0.5 13 0.4 33 ul:
Total continuum (including cusp)
0.1 3.2 3.2 2.8 2.8 7 7
1 no estimate bbe ed 458) | 16 12 4
4 23 0.5 20 0.4 50 1

243

TABLE VI. Spectra of phase difference 64 and phase 4 in cycles? per band (band width is 0.072 cpd=2/month). The spectra of
phase difference are given separately for the first and second fortnight.

Bermuda Midstation
5b & 5b $

epd 1st. 2nd Comb. Comb. 1st 2nd Comb. Comb.
Subinertials
0.07 0.05 1078 0-10 1078 0.07x 107° 54.38 0.07x 1073 0.011079 0.04x10° 46.65
0.14 0.08 0.05 0.07 7,40 0.04 0.00 0.02 14.81
0.21 0.07 0.00 0.04 1.86 0.04 0.01 0.02 6.83
0.29 0.03 0.04 0.03 3.94 0.01 0.01 0.01 4,29
0.36 0.06 0.12 0.09 POH 0.08 0.00 0.04 as
0.43 0.02 0.08 0.05 1.49 0.07 0.01 0.04 ae
0.50 0.09 0.01 0.05 0.99 0.01 0,01 0.01 rai
0.57 0.01 0.04 0.02 0.69 0.01 0.02 0.02 z
0.64 0.06 0.02 0.04 0.27 00 0.02 0.01 0.26
0.72 0.11 0.04 0.08 0.77 01 0.01 0.01 0.45
0.79 0.06 0.00 0.03 0.32 202 0.03 0.02 0.35

0.57 x1078 74,32 0.24x10° 80.45
Diurnals
0.86 0.021078 0.111079 0.06 107° 0.29 0.01* 1078 0.001078 0.01™ 1079 0.30

: 0.93(0,) 0.07 0,03 0.05 0.15 0.05 0.00 0.03 0.30
Inertials ‘s

1.00(K;) 0.04 0.23 0.14 0.22 0.02 0.03 0.03 0.16
1.07 0.12 0.04 0.08 0.12 0.08 0.01 0.05 0.23
1.14 0.10 0.08 0.09 0.30 0.09 203 0.06 0.26
2 0.06 0.16 0.11 0.36 0.03 03 0.03 0.35
1,29 0.20 0.02 0.11 0.29 0.10 06 0.08 0.21
1.36 0.17 0.10 0.14 0.13 0.00 .02 0.01 0.12

0.78% 10° 1.86 0.30%10° 1.93
Semidiurnals
1.60 0.271078 0.13 1073 0.20107 0.28 0.031078 0.10 107° 0.061078 0.06
1.66 0.05 0.31 0.18 0.24 0.14 0.22 0.18 0.04
TLS) 0.81 0.40 0.61 0.34 0.04 0.05 0.04 0.20
1.80 0.23 0.21 0.22 0.25 0.01 0.29 Owls 0.14
1. 86 3.44 1.83 2.63 1.84 0.13 0.89 0.51 0.16
1.93(M)) 0.87 1.91 1.39 0.86 3.08 1.62 2.35 1.64
2..00(S.) 1.88 0.17 102 0.53 0.64 Bal2 1.38 0.79
2.06 0.71 0.42 0.57 0.32 0.01 0.24 0.13 0.24
2,13 0.19 0.30 0. 24 0.17 0.01 0.06 0.03 0.03
2.19 0.44 0.13 0.29 0.08 0.01 0.06 0.04 0.10

7,35%10° 4,91 4,87x10% 3.40

Total Variance, !° all frequencies

Fortnight 60x 1075 60x 107° 60x 1075 90 31x 1073 27x 1078 29x 1078 98
Total 60x 1078 110 291078 548

Rei =) R,ei%its®) = 18°F Rieiti, j=y=1 , (44)
i i

and so the multipath phase is rotated by the single-path
phase shift A®. The measured 2.5 square cycles in

the semidiurnal tidal peak of multipath phase corre-
sponds to an amplitude of /2X2.5=2.2 cycles, in close
agreement with the computed 2.3 cycles for a typical
tidal current. But for the Cartesian multipath, the tidal
energy peak is reduced by the number of paths (n= 34
for Bermuda) and should no longer be discernible. So
the hypothesis is in very satisfactory agreement with
observations.

But it turns out that the MIMI propagation path runs
through the MODE expedition area, ‘ the only place

where a grid of deep-sea tidal pressure measurements
have ever been taken (Fig. 11). Currents can be com-
puted from the pressure gradients. M, tidal currents
have amplitude close to 1 cm/sec, but the MIMI path is
almost at right angles to the major axis of the tidal el-
lipse, and the MIMI component is small and poorly de-
termined (Table VIII). (A computer model of tides”
has the minor axis in opposite phase.) We expect the
tidal phase (Greenwich epoch °G) of maximum current
towards Bermuda to coincide with minimum acoustic
phase; i.e., 180°+°G for &(t) should be about 180° to
agree with MODE measurements. In fact, phases vary
from fortnight to fortnight; there is no resemblance be-
tween midstation and Bermuda.

The variability from fortnight to fortnight of a

TABLE VII. Power per band (bandwidth is 0.072 cpd=2/month of Fy(w)+Fy(w) for the first and second fortnight, and for the
combined record.

EEE

Bermuda Midstation
cpd 1st 2nd Comb. 1st 2nd Comb.
Subinertials
0.07 0.34x 101! 0. 24x 101! 0.29x 101 0.51~ 10!* 0.14 10” 0.32x10!?
0.14 0.12 0.22 0.17 0.12 0.60 0.36
0.21 0,22 0.07 (0,25 0.65 0.43 0.54
0.29 0.12 0.08 0.10 0.10 0,24 (pay?
0.36 0,02 0.17 0.08 0.43 0.07 0.24
0.43 0.23 0.19 0,21 0.06 0.37 0,22
0.50 0.12 0.04 0.08 0.46 0.10 0.28
0.57 0.08 0.18 0.13 0.37 0.30 0.33
0.64 0.17 0.16 0.16 0.16 0.29 0.22
0.72 0,15 0.04 0,09 0.08 0.17 0.13
0.79 0.13 0.53 0.33 0.36 0.44 0.40
SY AY ion 1.80 3,28 3.17 3,23
Diurnals
0.86 0.27% 101! e240! 0.25% 104 0.2410! 0.1510! 0. 20x 101?
Inertial ~0.93(0;) 0.06 .16 0.11 0.57 0,10 0.34
at 1. 00(K;) 0.08 seul 0.15 0.15 0.72 0.44
1.07 0.29 12 Oa 20 0.82 0.47 0.64
1,14 0.04 03 0.04 0.24 0.20 0.22
L 2k 0.17 07 0.12 0.36 0.64 0.50
1,29 0.10 malat 0.10 0.26 0.10 0.18
1.36 0.36 08 0,22 0.13 0.25 0.19
CY 1.02 1.19 2.78 2.63 2.71
Semidiurnals
1.60 0.32x10!! 0.14101! 0.23104 0.22x 101? 0.06 101° 0.14x10!?
1,66 0.13 0,12 0.13 0.07 0.19 0.13
1.73 0.21 0,16 0.18 0.24 0.09 0.16
1.80 0.04 0.08 0.06 0.27 0.05 0.16
1.86 0.23 0.10 0,17 0.42 0.09 0.25
1.93 (M,) 0.01 0,22 0.11 0.43 0.04 0.23
2.00 (So) 0.07 0.14 0.10 0.39 0,19 0.29
2.06 0.19 0.01 0.10 Witsts) 0.30 0.59
2,13 0,29 0.04 0.11 0.13 0.05 0,09
Zid 0.18 0.13 0.16 0.39 0.11 0.25
1,57 1,14 1.35 3.45 1.16 2.31
Total variance, all frequencies
Fortnightly 132x101! 174x 10!”
broadened semidiurnal phase-peak is in line with the This suggests that the location of the acoustic source
known character of internal tides. (At Bermuda the and receiver in the generating area of internal tides
spectral peak occurs at one harmonic below M, fre- may be a Significant factor.

quency.) Internal tides have wavelengths short com-
pared to the acoustic paths, and one would expect them
to produce an incoherent phase modulation. An excep-

F . é ; TABLE VIII. M, component of tidal current and acoustic
tion might be the terminal effects. Internal tides are

generated by conversion from surface to internal modes aaa
in regions of prominent bottom topography, which is Tidal current, azimuth 55° Amplitude °G
just where the hydrophones are located. In such re- MODE GNoReGhononte Onstcmn/scommao
gions the internal tides may dominate, whereas in the DarlkeeHonderehotimodel 0.5 20
open sea internal tides typically have 10% of the internal
wave energy. Take a large vertical tidal displacement Acoustic phase &(t) Amplitude 180°+°G
¢=10 m, corresponding to 6C/C =10"* at 1-km depth; TWiastion datiortaight === @eyeles, 0760,
then in a near zone of radius R=)/27, with A=100 km 2nd fortnight 1.1 260
for the wavelength of the lowest internal tide mode, we '
neve Bermuda Ist fortnight 0.3 105

2nd fortnight 1.3 179

AG = 270(R/C) (5C/C) = 2.6 cycles.

90°W 60°

of i
aN $
.
a
Ss se =
aN fs
\ Ro) “3
aN
ot ‘
W\Y
Me,
PX)
a
0 1 2
cm/sec

Finally, there is the possibility of incoherent phase
modulation by internal tides along the entire path. This
is then analogous to the incoherent modulation by inter-
nal waves in general. Internal tides have typically one-
third the amplitude of the internal waves (Table V), and
so contribute 10% to (62). If this were the only contri-
bution, then because of the periodic input at tidal fre-
quencies w,, the multipath spectra would be concen-
trated at w,, 2W,, 3w,, ..., and the local energy density
would be high. In the presence of internal waves there
is interaction with all frequencies, but some remnant of
the tidal line spectrum can be expected to remain. The
problem needs further consideration.

ACKNOWLEDGMENTS

John Clark and his associates have furnished the
acoustic records on which this analysis is based.
Flicki Dormer and Betty Ma have carried out the data
reduction. Discussions with Ted Birdsall have been
most helpful.

APPENDIX A.
Equation (10) can be written
e n
G2 =v?=v5 [ widw=vélns ,
Win
where s=n/w;>1. Hence
cr) n
G=v0f wdy ~$n* v2
Win

and

OD. at -20/b
a? Op liswee ae

for n=nge*/, ng=5.2X10° sect, v?=3.2x10° sec®,

APPENDIX B.
In some oceanic models we may have a relation

(64 ,=0(63),? , (9’)

246

FIG. 11.

Greenwich epochs.

My tidal currents in MODE
area between Eleuthera and Bermuda,
The arrow toward 0°G refers to the
current vector (scale below) when the
Moon passes over the Greenwich merid-
ian; 30°G, 60°G,..., refers to other
The upper ellipse
refers to a computer model by Parke
and Hendershott, the lower ellipse is
based on deep-sea tide measurements.

with a coefficient f replacing the 3 which appears in
Eq. (9). In this case the following changes need to be

made in the results of Secs. Il and III:
yer iey
4 fe? <1
to=filey P
Q(m) = (20?) £32
(ag)=f7? ,

(Op )) = (0/3) £1 /? + (02/8) 1/2 (aft /? — 1)

a=(n/3) + (12/8) (nft/? —1)4

(26’)
(27’)
(28’)
(29)
(30’)
(33’)
(34’)

For example, if f=9, a=1.19 and c=7/2a=1.32 in

Eq. (40’).

APPENDIX C.

The mean-square vertical displacement is

(r= sete 2 (7 eulet awh

mM Ny in fl

where
ed, (72492) 20.468 «
j?
The relative contribution
w
[ood
ini
equals
§ —(V3/27)=0.39, from w,, to 2w,,
=1, fromw;, to ,
whereas the j contributions are
0.214, 0.164, 1, for j=1, j=2,), .
gal

Similarly,

given by

(7 +98)?
af ’

Shes Se ear airare) 5 2492)
Bees — Sylow dl AFBI y/o
(2) 5 B Brg | af ost on) wd, 7
eee + a =0.76, from to 2
3 oF Sey y Win Win

; from w,, to ©

with the j contributions as before.

*The work on random vector statistics was started during the
1974 JASON Summer Study under the auspices of Stanford
Research Institute, supported by the Advanced Research
Projects Agency. Subsequent analysis has been supported by
the Office of Naval Research.

Thor Miami—Michigan project, starting with J. C. Steinberg
and T. G. Birdsall, “Underwater Sound Propagation in the

Straits of Florida,” J. Acoust. Soc. Am. 39, 301—315 (1966).

For a recent paper see J. G. Clark and M. Kronengold,
“Long-period fluctuations of CW signals in deep and shallow
water,’ J. Acoust. Soc. Am. 56, 1071-1083 (1974).

23. G. Clark, “Ray Propagation in an Underwater Acoustic
Channel with Time Varying Stratification,’ Tech. Rep. ML
70107 (University of Miami, Rosenstiel School of Marine and
Atmospheric Science, 1970); I. Dyer, ‘Statistics of Sound
Propagation in the Ocean,’ J. Acoust. Soc. Am. 48, 337—
345 (1970); J. G. Clark, N. L. Weinberg, and M. J. Jacob-
son, “Refracted, Bottom-Reflected Ray Propagation in a
Channel with Time-Dependent Linear Stratification, ” J.
Acoust. Soc. Am. 538, 802—818 (1973). H. A. DeFerrari,
“Effects of horizontally varying internal wavefields on multi-
path interference for propagation through the deep sound
channel,” J. Acoust. Soc. Am. 56, 40-46 (1974). H.
DeFerrari and R. Leung, “Spectrum of phase fluctuations
caused by multipath interference,” J. Acoust, Soc. Am, 58,
604=607 (1975).

5Surface scattered arrivals differ in frequency from o by
roughly +0.1 Hz (the frequency of ocean waves) and are re-
moved by narrow-band filtering. Bottow-scattered arrivals
are greatly attenuated.

‘In fact the phases were computed 32 times for each 5 min
period and then averaged [T. Birdsall (private communica-
tion)]; this may account for the reasonable behavior of phase
spectra (as compared to intensity spectra) at the high fre-
quency limit. It is the reason for the phase jumps between
5 min readings (Sec. I).

5c. J. R. Garrett and W. H. Munk, “Space-Time Scales of

Internal Waves; A Progress Report,’ J. Geophys. Res. 80,
291-297 (1975). J. L. Cairns and G. O. Williams, “Inter-
nal Wave Observations from a Midwater Float: Part II,”

J. Geophys. Res. (in press) (1976).

‘W.H. Munk and F. Zachariasen, “Sound Propagation Through
a Fluctuating Stratified Ocean; Theory and Observation,”

J. Acoust. Soc, Am. (in press) (1976).

"Units are a bother, but the usual way out of plotting loga-
rithmically and labeling decibels won’t do. F3(w) has the
dimension of frequency, and the spectrum is a plot of fre-
quency versus frequency, derived as follows: In Eq. (41)
and (43), both ¢ and w are in radians per second, and F3(w)
gives the contribution, per unit band (rps), to (ob?) in (rps)’,
hence Fj(w) has the dimensions (rps)*/rps=rps. The mea-
sured time series is 64: the phase difference (in cycles)
during an interval 6¢. The contributions to ((6)*) are dis-
tributed among 64 frequency bands between 0 and (26¢)"},
each of width (128 6t)"! cps=27(128 5t)"! rps. The plotted
spectrum is then

Pep) = 9 (61)" 2n(128 61)! F3(w)

1/2
_ vét ( P @)
Sogn Se ,

with v in sec™!, 6f=300 sec, and w(Hz) =(27/86400) w(epd).
The plotted intensity spectrum is

10 \° 2n
In a) Toset Fs) -

Fya)=(
8N. L. Weinberg, J. Clark, and R. P. Flanagan, “Internal
tidal influence on deep-ocean acoustic-ray propagation, ” J.
Acoust. Soc. Am. 56, 447-458 (1974).

°L. Magaard and W. D. McKee, “Semi-diurnal Tidal Currents
at ‘Site D,’”’ Deep-Sea Res. 20, 997-1009 (1973). C.
Wunsch, ‘Internal Tides in the Ocean,” Rev. Geophys.
Space Phys. 13, 167—182 (1975). C. N. K. Mooers and D.
A. Brooks, “Tidal and Longer Period Fluctuations of Inter-
nal and External Fields in the Florida Current, Summer
1970,” Deep-Sea Res. (in press) (1976). M. G. Briscoe,
“Preliminary Results from the Tri-moored Internal Wave
Experiment (IWEX),” J. Geophys. Res. 80, 3872-3884
(1975).

For midstation, the total (#2) greatly exceeds both fort-
nightly (®*), see Fig. 4.

‘'B, Zetler, W. Munk, H. Mofjeld, W. Brown, and F.
Dormer, ‘‘MODE tides,” J. Phys. Oceanogr. 5, 430—441
(1975).

yt, Parke and M. Hendershott (personal communication) ,

247

ra.

—

ACOUSTIC PROPERTIES OF THE SEA FLOOR

John Ewing

Lamont-Doherty Geological Observatory
Columbia University
Palisades, New York

Studies of the reflection and refraction of sound by the
ocean bottom and sub-bottom have provided the basis for
characterizing geographical regions (provinces) in terms

of sound velocity versus depth functions. Velocity gradi-
ents vary appreciably from province to province in response
to variations in sediment type and in mode and rate of
deposition. When the gradient is expressed as V = V_ + KT
(where T is one-way travel time) the value of K geneYally
lies between 0.5 and 1 sec ~. These values represent
average gradients in 1 km or more of sedimentary section.

Recent analysis of two data sets from the Hatteras abyssal
plain has provided an opportunity to examine local varia-
tions in the velocity versus depth function and to investi-
gate energy distribution among the various reflected and
refracted paths. The region can be characterized reason-
ably well by two linear velocity versus depth functions:

vV = 1.5 + 2T for the upper 400-500 meters and V=1.9+T
for the lower part of the sedimentary section. Standard
deviations of sound velocity in the two data sets are be-
tween 50 and 100 meters per second.

For frequencies in the range of 60 Hz and lower, the signal
amplitudes associated with rays penetrating to a reflector
500 meters deep in the sediments are, in a substantial range
of grazing angles, 6 to 10 decibels higher than amplitudes
associated with rays reflecting from the sea floor and
shallow interfaces. Comparably high signal amplitudes are
received at discreet ranges from still deeper levels, down
to the top of the igneous basement. Some variations in
signal level can be related to multipath interference and
geometrical focusing effects, as well as to change in co-
efficient of reflectivity associated with incident angle.

The following is a brief summary of work we have been doing in

the marine seismology group at Lamont on ocean bottom acoustics. We

249

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

have been looking at bottom effects through the use of expendable
sonobuoys and air gun sound sources and have recorded a large number
of variable-angle reflection profiles. Since our ships travel around
the earth widely, this has given us a chance to look at the bottom of
the ocean in many places and to compare the behavior in one place

with that in another.

Although not completely satisfied with the quality of each of
our measurements, I do think that our total data bank is starting
to give meaningful information about what is going on, particularly

at low frequencies, when sound encounters the bottom.

Figure 1 is an example of an airgun-sonobuoy reflection pro-
file, the ordinate representing reflection time and the abscissa
representing distance. At these low frequencies (about 20 to 60 Hz),
several reflectors appear quite clearly as a distinct reflection
hyperbolic curve. The sea floor reflection intercepts the ordinate
at about 6.2 seconds and prominent reflections from within the
sedimentary section have intercepts of about 7.2 and 8.2 seconds.

The intercept at about 9 seconds corresponds to igneous basement.

It is quite clear at ranges between about 12 and 16 km (corresponding
to grazing angles on the bottom of 30 degrees or so) that a lot of
these reflection curves are starting to run together. When they do,
we get some interference patterns showing up and the signal levels
observed in that part of the profile vary extremely widely over

(I would guess) something like 20 dB.

Even before we get to these moderately small grazing angles,
at frequencies in the vicinity of 20 Hz, we are already getting a
lot of energy from the sub-bottom interfaces. By the time the grazing
angle reaches 45 degrees, in many places we get at least as much low-

frequency energy from reflectors at depths of 500 meters or more as

250

ACOUSTIC PROPERTIES OF THE SEA FLOOR

EWING

SAIN YS

wade

auOsS pue STPATIIe (aAeK
ay WOLF SAAEM PADIT FAI pue (S9UTT WYSTeIS) 9AVM JISLTP adeFins Burmoys

a ee ee ee es

dIIdOud AONAONOS - NNOYIYW “T eanbty

‘weisetTp ay} FO WYITA LOMOT ut ivedde sTeATIIe PpseYdaTjfei a[Bue-aptm
pesy) paqderjyay ‘(saaino DT TOqIacdy) sSadeF1ozUT wo OG-qns pue IOOTF vas
STTFOL -undity

an Steer. S . . oe Bes

e OSDP Value

ZO km.

~
Ni

Wieck ears

251

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

we get from the sea bottom. That says at least two things to me:
That many sub-bottom interfaces have rather high reflectivity and,
probably to nobody's surprise, that the attenuation in these moder-
ately soft sediments is certainly not very high for frequencies in

the 20 Hz range.

We see this low attenuation demonstrated in a slightly different
way as we travel along almost any ocean basin where the igneous rock
surface of the earth's crust is covered by a variable thickness of
sediment. As you cross such a bottom and make a low-frequency echo
sounding record, you can see little difference in the intensity of
the reflections from the basement surface whether it is covered by

a few tens of meters or a few hundreds of meters of sediment.

We have recently made several airgun-sonobuoy measurements in
the Hatteras abyssal plain in connection with some joint work with
NUSC. I was particularly interested in the Hatteras abyssal plain
because I remember from some of the early work in bottom loss
measurements made at 3.5 kHz that the Hatteras abyssal plain was con-
sidered to be about as good a reflecting bottom as we knew. We knew
from piston coring and some of the Glomar Challenger work that this
abyssal plain had quite a lot of sand and silt in it, so it ought to
be a good reflecting bottom. We also knew there was another reflector
about 500 meters below bottom, one that we observed very broadly over
the North Atlantic Ocean, and we knew that it corresponded to some
closely spaced layers of chert (flint) in otherwise soft sediment.

I was quite interested to see how the reflectivity of the sea floor
in this nice, smooth abyssal plain would compare with the reflector
about 500 meters below the sea floor that I knew had some fairly

hard rock associated with it.

252

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

Sure enough, in all of the profiles there, by the time we were
out to grazing angles of 30 degrees, we consistently had signal
levels coming from this subsurface reflector anywhere from 6 to 10 dB
higher than signal levels from the sea bottom. In some places,
usually at grazing angles of 30 to 45 degrees, it was common for the
largest signal received in any part of the signal train to be coming
from even deeper than the 500-meter level, sometimes coming from the

top of the igneous rock itself, 1,000 meters or more below bottom.

One thing more. Notice in Figure 1 that at these farther ranges
some signals are arriving appreciably ahead of the reflected signals.
These are head waves coming from some of the deeper, high-velocity
layers. Although they are interesting and important to us in geo-
physics, they do not carry much energy. They may appear to be rather
energetic in the figure, but that is because this particular buoy is

an SSQ41 buoy with AGC.

To summarize this part of my talk — there are large areas of
the sea floor where, at frequencies below 100 Hz, appreciably more
energy is returned to the surface by reflection from interfaces well
below the bottom (hundreds of meters) than is returned from the sea

floor itself.

We also get velocity information from the airgun-sonobuoy pro-
files. The technique that we have been using is rather standard,
developed for geophysicists by Dix many years agc. It is known as the
x? - 7p? method and is a purely geometrical treatment of the problem
that depends on the fact that the shot point and receiving point

separate during the experiment.

These measurements are easy to make. From them we can calculate

interval velocities for each layer that is bounded by distinct

253

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

interfaces. If in one province we get enough such measurements, we
can construct a plot of interval velocity versus depth in the sedi-

mentary column.

We usually plot the depth function in one-way travel time, in
seconds. The inset in Figure 1 shows a velocity/depth plot for the
western North Atlantic rise. Here, as in most other places, we find
that we can fit these data with a function that is linear in time
although not quite linear in depth. If we express V = Nes + kT, where
T is one-way travel time vertically through the sedimentary section,
k in this equation is in units of jen/see Most physicists, I think,
tend to think of sound-speed gradients in depth rather than time,
which are typically expressed in terms of kilometers per second per
kilometer or just in Seconcemae In most of the velocity range that
we are dealing with in soft marine sediments, these two types of
gradients turn out to be only about a factor of two apart. In other
words, a gradient of about one per second corresponds to a k of

approximately two kilometers per second per second.

In a lot of our measurements of this type from around the world,
we characterize different areas in terms of this value, k, which, in
fact, is characterizing the sea bottom in terms of velocity gradient
in the sediment. Before I summarize these measurements, refer again
to the inset in Figure 1. I pointed out that there is quite a lot
of scatter in these data. The reason we can get this many data
points is that the geology changes even in a rather local region. At
one place we may see a reflector at some depth below the sea bottom;
in other places we may be measuring it at half that depth or twice
that depth. So if we make enough measurements, we get a fairly good
distribution of layer thicknesses,and, therefore, we get several
values of T. For each value of T, which is a measure of depth in the
section, we calculate interval velocity so we can get a good distribu-

tion of velocities versus depth.

254

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

In the areas that we characterize by certain values of k, we
notice that typically the standard deviation in velocity determina-
tion is about 100 meters per second. I have not been satisfied with
that value; I thought we could probably do better. One of the
reasons we wanted to do the set of measurements in the Hatteras
abyssal plain was it gave us a chance to go to a localized area and
do several of these experiments to see how much the scatter in deter-
mining velocity resulted from real geological change and how much,

perhaps, resulted from some shortcoming in our method.

It turned out that in the closely grouped measurements in the
Hatteras abyssal plain the velocity scatter did not appreciably change
over what we had derived from 30 or 40 measurements over the whole
Hatteras abyssal plain and part of the lower continental rise. This
result caused us to consider whether our treatment of these data SES
paying enough attention to the details of the structure in the water

column.

We had initially treated the water column in the x? - 7 calcu-
lations as though it were a constant velocity layer, figuring that
we were working mainly with rather steep ray paths for which the
constant velocity assumption should produce only a small error. In
our first attempt to improve this model, we divided the water layer
into several layers, but this did not seem to reduce the scatter in
the velocity versus depth determinations. A better water model
shifted the average somewhat, not surprisingly, but it didn't really

take the scatter out of these data.

A scheme proposed by George Bryan and representing an effort
to escape the water layer model is demonstrated in Figure 2. It is
a very simple two-layer model, water and sediment with a reflector
at the bottom of each layer. The reflection curves and ray paths are

shown.

255

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

Figure 2. DIAGRAM (FROM BRYAN, 1974) SHOWING RELATIONSHIP
OF BOTTOM-REFLECTED AND SUB-BOTTOM-REFLECTED
RAYS WITH DERIVATIVES OF TIME VERSUS DISTANCE

CURVES

256

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

As you can see from the model, it is possible to take pairs of
rays in which the rays from the bottom reflection and from the sub-
bottom reflection are parallel to each other in the water column. If
we differentiate these reflection curves, the derivative tells us
the slope of the curve, of course, and, physically, the inclination
of the ray at the sea surface at that point. In effect, if we go
along these reflection curves and find pairs of derivatives that are
the same, we are finding pairs of rays (of which one is a bottom
reflection and one is a sub-bottom reflection) that have traveled
parallel, and presumably equal, time paths through the water. Thus,
each pair of common derivatives gives us a AX and a AT associated
with the path through the sediment layer, as shown in the diagram.
We then carry out this procedure over a wide range of AXs and ATs,
plot an x? - 0 profile and get a value of interval velocity for the

sediment layer.

We treated a substantial amount of our data in this way and we
still have a lot of scatter — more than I like. This treatment
should take account of the water structure, but, of course, it only
takes account of a fixed water structure. No matter how you analyze
these data, the water layer is a part of the model and if it changes
significantly during the course of the experiment, you still have

a problem.

We plan to put our entire experiment on the bottom of the ocean
as one way to answer the question for certain whether our scatter in
velocities versus depth results from the water column or from geology.
I'd be very surprised and disappointed if there were no geological
effect. But I have yet to be convinced that all of the variations

are geological ones.

257

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

Figure 3 shows the distribution of our sonobuoy wide-angle
reflection measurements on a world basis. The boxes indicate areas
where we seem to have enough measurements in a geologically definable
province to characterize it. Figure 4 shows a valne of k (gradient)

for each of these same areas.

Do not pay any attention to the central equatorial Pacific area.
It indicates a very high value of gradient with a k value of 3.9.
Although the value is correct, it represents a special case of some
very thin, low-velocity sediments on top and some very high-velocity
limestone at the bottom. It more properly ought to be treated as
a two-layer case. The other numbers are the best values we can pro-
duce at present. Remember that the numbers represent k in the linear

expression V = Nis adie

Our methods of measuring from the surface are just not good
enough to determine with precision the uppermost sediment velocity
(< 100 meters thickness), but some characteristics of these data
give us very good reason to believe that in the uppermost 100 meters
or so is a considerably steeper gradient than the value listed for

the entire section.

I want to discuss now the distribution of sediments. This is
important because if negative bottom loss is a reality, it is be-
cause velocity gradients (and good sub-bottom reflectors) form, in
effect, an acoustic lens at certain ranges. The more sediment we
have, the more possibilities we have for acoustic lenses of various
characteristics, to say nothing of the smoothing effects of sub-
stantial thicknesses of sediments. So it is of some interest to us,

I think, to know the distribution of sediments around the world.

Figure 5 gives the distribution for the Atlantic. Although you

cannot see the thickness contours, you can see the hatched region in

258

ACOUSTIC PROPERTIES OF THE SEA FLOOR

EWING:

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SNOTLONNaA HLddd SOSHHA ALIOOTHA ANNOS LNAYWIGHS YVIINIS ONILIGIHXd
SVduYVY GHYNITLNO HLIM SNOILVLS AONHONOS-NNSYIV AO NOTLNAINLSIAG

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259

ACOUSTIC PROPERTIES OF THE SEA FLOOR

EWING:

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260

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

Figure 5. SEDIMENT DISTRIBUTION IN THE ATLANTIC OCEAN

261

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

the middle of the ocean where there are less than 100 meters of sedi-
ment. The stippled area on either side has more than a kilometer of
sediment. Close to the continental margins on either side are as

much as 5 or 6 kilometers of sediment.

Figure 6 is a similar display for the North Pacific Ocean. In
this area, and in several other localized areas, we now have more
detailed charts but this shows the general distribution. As in the
Atlantic, we find little sediment, less than 100 meters, in broad
areas. Sediments in the northeastern part are much thicker because
of a great addition of turbidite sediments (detrital sediments).
There is a nice thick belt of sediments along the equator caused by
upwelling of deep water and high biogenic productivity. In the
western Pacific, we find a distinct case of a two-layer situation
with a thin layer of soft sediment overlying a much thicker layer of

very hard sediment, the hard sediment being a cherty limestone.

Figure 7 shows the results of one of the JOIDES holes in the
western Pacific where we paid particularly close attention to several
factors. The reflection profile is traced on the right and next to
it are shown interval velocities. The lithologic section that was
cored is in the middle. This is a hole about 1,200 meters deep.

The heavy trace on the left is a plot of the age of the sediment
down the hole in millions of years. The dashed trace is the drilling

record in terms of drilling time in minutes per meter.

We got nice correlations in the drilling record with the reflec-
tors at about 600 meters and 800 meters. Most of the upper 600 meters
of the section is just ooze, a microfossil ooze. At 600 meters, an
exceedingly sharp interface occurs where chert (flint) layers have
developed. This interface represents the rather abrupt transition
between soft sediment, that you can make a mud ball out of, and these

very hard chert layers that you can make arrowheads out of.

262

ACOUSTIC PROPERTIES OF THE SEA FLOOR

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263

Meters

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Drilling Tine,

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EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

6 4 2

min./meter

<r,

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SITE 167

Average
Velocity,
km/sec

Middle focene -

Late Cretaceous

Vw
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COMPARISON OF SEDIMENT AGE, DRILLING RATE,
STRATIGRAPHIC SECTION, SEISMIC PROFILER RECORD
AND COMPUTED INTERVAL VELOCITIES FOR DSDP

Sire, 167

264

Seconds

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

Below the 600-meter interface is reasonably firm sediment down
to about 800 meters where calcium carbonate chalk turns to limestone,
really hard limestone. This transition also produces a very good
reflection and a very sharp drilling break. At 1,200 meters we hit

basalt under the limestone.

We now have several such holes from the JOIDES program that are
well enough cored and geophysically examined so that we are starting
to know what causes a lot of our reflectors. I think you can under-
stand that it is not only interesting to us in a geological sense to
identify the reflectors, but that the identification also permits us
to use geological reasoning to interpolate between data points and

gradually to build up a more complete geoacoustic model.

Figure 8, a section based on seismic data and drilling in the
Atlantic, is the southern part of the Hatteras abyssal plain. We
have identified some friendly Atlantic reflectors here. We have an
interface in the sediments, fairly shallow in some places, deeper in
others, labeled "A" which we now know is a series of chert beds,
nearly the same age as those in the Pacific. A thick layer of clay
is underneath, then again nice hard limestones (8) near the base of

the section, and then the basalt (B).

REFERENCES
Dix, C. H., Geophysics, 20:68-86 (1955)

Bryan, G. M., "Sonobuoy Measurements in Thin Layers," in Physics of
Sound in Marine Sediments, L. Hampton, ed., Plenum Press (1974)

265

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

\ i

Figure 8. BLOCK DIAGRAM SHOWING PRINCIPAL REFLECTORS AND
SEDIMENT LITHOSTRATIGRAPHIC UNITS IN THE WESTERN
PART OF THE NORTH ATLANTIC

266

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR
DISCUSSION

DR. W. H. MUNK (Institute of Geophysics and Planetary Physics,
University of California at San Diego): Is the scatter on Figure 1
oceanographic or geological? Also, are there any measurements on
land in sediments that could give a clue as to whether the order of

scattering there is consistent or not with geologic inhomogeneities?

MR. EWING: We have some data that really made me suspect most
strongly that it was the water column that was causing this. For
example, in the Hatteras abyssal plain which our seismic data indi-
cates to be a nicely layered section of sediment, the individual
reflectors can be followed for hundreds of miles. The bottom seems
to be just a beautiful cake of sediment. The data scatter represents

a standard deviation of a hundred meters per second.

We can move up onto the continental shelf where from a geologi-
cal point of view I would expect a bigger variation in geology, and
there we get maybe half of that standard deviation. I think that is
because we removed a lot of the water problem by going to shallow

water.

DR. H. WEINBERG (New London Laboratory, Naval Underwater Systems
Center): It seems to me that you are using ray theory at low fre-
quencies and shallow grazing angles, and we have seen that this is
one case when ray theory can really get you into a lot of trouble.
Have you every tried to incorporate a better theory than regular ray
theory; what would happen if you treated the propagation loss

directly?

MR. EWING: We are concentrating primarily at the moment on
developing the best model we can for velocity gradient. Working at

appropriate incidence angles keeps you away from ray theory problems.

267

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

DR. WEINBERG: I found that the grazing angle can be 10 degrees.
For example, if you go to the extreme case, the ray that grazes the
ocean bottom has an infinite propagation loss, and that would tell

you that you had a negative infinity bottom loss.

DR. M. SCHULKIN (Naval Oceanographic Office): If you
consider the bottom in terms of velocity gradient and absorption,
that is, consider it was an extension of the water medium, you do get
convergence zone type propagation from very steep velocity gradients.
It is possible to get an effect of negative bottom loss in the first
bottom bounce region. Of course, beyond that it goes off as 3 dB per
distance doubled as far as the loss goes. So that you only get this
apparent gain in that first zone. But the rays penetrate the bottom
and you just carry the ray tracing procedure through with the

correction for the convergence effects.

DR. WEINBERG: That is a possible explanation, but there is
another one. If you just take the velocity gradient in deep water
and you have a positive velocity gradient going down, instead of
using plane waves use Airy function solutions and you may do away

with the negative bottom loss.

MR. W. H. GEDDES (Naval Oceanographic Office): There are a
number of alternative explanations. I wouldn't want to hold out for
the ray trace solution without saying that the negative loss is a
flag indicating that the model (used in this way) is going to produce
some strange answers. What I really want to hold out for is an
appreciable amount of energy being refracted through the bottom and
that it may not be a reflection arrival at all. I don't hold for

the negative losses is what I'm saying.

268

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

DR. J. S. HANNA (Office of Naval Research, AESD): I didn't want
to say too much right at this point about these problems with negative
reflectivity because I had some comments I was expecting to give this
evening in my discussion which are germane to some of the shortcomings
or deficiencies in the transmission loss model used to reduce these

data.

There are several effects one needs to worry about — the kinds
that were mentioned here earlier as well as the implications of third-
octave band processing with regard to whether you are adding these

arrivals coherently or incoherently.

The particular model that was used here assumed that the four
arrivals added without regard to phase. This is not strictly speaking

true at low frequencies with third-octave processing.

DR. S. M. FLATTE (University of California, College at Santa
Cruz): I wanted to ask Ewing a question. When you are comparing two
paths where you try to cancel out the effect of the water column,
there are of necessity still two paths which go through different
parts of the water column. What is the typical difference in travel
time that would have been assumed equal that would cause your

scatter in points on the velocity determination?

DR. EWING: Which are the other two paths, Stan, that you are

talking about?

DR. FLATTE: The direct path goes through a different part of

the water than the one which has traversed the bottom layer.

MR. EWING: It goes through a different part of the water, yes,
and our only assumption was that if there is no horizontal variation,
then we should have eliminated most of the problems with the water

column. The fact that we did not eliminate most of the problems led

269

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

us to suspect that we do have to worry quite a lot more about the

horiziontal changes.

DR. FLATTE: Right. And my question is a quantitative one.
What horizontal difference does there have to be in order for you to

get the scatter you observed?

MR. EWING: I'm not sure I can answer you without a little

thinking.
DR. FLATTE: If it's a hundred meters — If it's a fifteenth of
a second — I'm not really sure that it is though because you have to

determine velocity and depth of layer at the same time. But if it is
a fifteenth of a second — what model could you make of the water
column that would do that? Because internal waves can't do it I'm

-4
sure, at the expected level of 10 for 6c/c.

MR. EWING: It does not take a very big change. You see, the
derivative of the reflection curves gives us the angle of the ray at
the sea surface. If this ray has encountered very much of a perturba-
tion anywhere near the surface it works on an awful long lever to
change the angle of incidence on the bottom, and the angle of inci-
dence on the bottom in our kind of analysis is very critical. A
rather small angle change near the surface makes a big change in

AX. versus AT in the bottom layer.

DR. FLATTE: Might it be milliseconds' difference in travel time
that could be the effect?

MR. EWING: It's more the effect of changing the direction of

the ray, of course, than it is of anything else.
DR. FLATTE: Yes, but your experimental data are just travel time.

MR. EWING: Yes.

270

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

DR. FLATTE: And so the question is whether it is difference in

travel time you might have observed.

MR. EWING: I would need to do a little arithmetic before I

could answer you for sure. I don't know what the scale is.

MR. R. L. MARTIN (New London Laboratory, Naval Underwater
Systems Center): Santanello and Berstein at NUSC have also done
several measurements of bottom loss, and they have observed this
negative bottom loss below 10 degrees grazing. They approach the
analysis quite differently. They took the broadband signal and
isolated the direct and the first bottom-reflected pulse, and then
ran the filtering after that; rather than taking the propagation
model over the entire path, they just took the differences in the
propagation over the path increment differences of the direct and

bottom-reflected arrivals.

I would guess that this illuminates two questions that arise
in processing these data and coming up with negative bottom loss.
One is sensitivity of it to the particular propagation model used,
and the other is the coherent effect through narrowband filters. So
negative bottom loss has been observed using different analysis

methods.

DR. A. O. SYKES (Office of Naval Research): Does sedi-

mentary ooze act more like a fluid or like a solid?
MR. EWING: More like a fluid.

DR. HANNA: Referring to the comment that was just made here by
Bob Martin, if you are taking the difference in transmission loss
along those two paths it still presumes that your model for trans-
mission loss in the water is sufficiently good to get both of those
right. If the path interacts with either boundary, there are still

the influences of caustic shadows on the field and things of that

ial

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

sort which seem to be present every time you get to the geometry

corresponding to low grazing angles.

So there are still possible complications I think that need to
be properly considered even if you tried to improve the experimental

range.

DR. J. L. WORZEL (Marine Science Institute Geophysics Laboratory) :
I think John Ewing's answer to Al Sykes' question needs a little mod-
ification. The oozes on the bottom act like a liquid when they have
high porosity, but as they get buried deeper the porosity is reduced

and then they no longer act like that.

DR. SCHULKIN: One of the questions is: What is the sound speed
and absorption as a function of porosity? Also, how does porosity
vary with depth beneath the surface of the bottom? When do shear

waves start in?

MR. EWING: Well, we know very low velocity shear waves can be

developed in very short sediments. We have observed them.

DR. HANNA: I have a question related back to the problem of
the scatter of the data you referred to. Just to make sure that I
didn't misinterpret some of the things that you said before, I would

like to go back and refresh myself.

I thought I understood you to say that if G is of the order 1

per second then K is of the order of 2?
MR. EWING: More or less, yes.

DR. HANNA: Then the question I have refers to the accuracy of
the resolution in time that you can achieve for the one-way travel

time.

272

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

If I understood you properly, I believe the records showed a
pass band from 20 to 40 Hertz or something of that sort. It would
seem to me that, just very crudely speaking, the time resolution
implied by that pass band might be of the order of tens of milli-
seconds. I wonder what possible influence that resolution might have
on the scatter of the data that you plotted here. Using your value
of 2 for K, this resolution would translate into something like 50

or 100 meters per second scatter.

MR. EWING: I guess the answer comes in two parts. How
accurately can we time an arrival? The question I guess then is
what does that arrival really mean particularly if you are ina
region where you are having an interference of two low frequency
signals? I completely agree that this is a possible source of our

problem.

The kind of data we are normally using, a reflection arrival
for instance, we usually just pick on the basis of like phase but
not precisely like phase. I mean whether it's positive or negative.
In very low frequency situations, of course, that gives potentially

a rise of big timing errors. I'm aware of that.

We are trying to stay with arrivals that are separated enough
in time. I guess another part of my sidestepping your direct question
is the answer I gave to Walter Munk. We do the same thing when we
work with the sediments on the continental shelf. We treat that
data in exactly the same way. Yet we get a much smaller distribution,

much tighter distribution.

In other words, if we go to a rather localized area and shoot a
dozen sonobuoys in this fashion and plot them up this way with diff-
erent filter settings, we can pick different levels in the sediment
usually because some level will be reflective for one frequency,

another level will be more reflective for another frequency.

EWING: ACOUSTIC PROPERTIES OF THE SEA FLOOR

So we divide up the sediment column and calculate one of
the regression curves. By doing this we get a lot of statistical
leverage in a shallow area. We always wind up with a much tighter

regression plot than we do in the deep water.

There is maybe one exception to that, and that is the Bering
Sea. There we have 50 or 60 measurements distributed sort of all
over the whole basin and they group in quite tightly around regression
curves. Whether it's because the Bering Sea is a little more stable

oceanographically, I don't know. We're still struggling.

MR. C. W. SPOFFORD (Office of Naval Research): On these
phase differences, John, is the bottom flat enough that you can con-
sider those two rays to be identical in the water column? That is,
one ray doesn't spend another 10 meters or so in the bottom in depth
which could give you some huge differences here I would think? Is

the bottom flat enough to ignore this effect?

MR. EWING: I think in the Hatteras abyssal plain it is. Those
abyssal plains are the flattest things known in nature as far as I
know. We cannot measure the slope with an echo sounding system that

measures to plus or minus a fathom.

DR. HERSEY: The grades are typically one in five thousand in

the central portion of abyssal plain.

274

THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

GC. W. Hoston, Sr.

Applied Research Laboratories
The University of Texas at Austin

R. J. Urick (1973) stressed the importance of sound trans-
mission through the ocean floor in the computation of re-
flection loss at the ocean bottom. Strong sub-bottom
reflecting layers are not necessary since the wave is
refracted upwards when there is a strong velocity gradient,
as in sedimentary layers with the properties described by
E. Hamilton (1974) for the abyssal plain in the northern
Pacific Ocean. The Green's function for a point source in
a liquid with a linear velocity gradient was derived by

C. L. Pekeris (1946) and D. H. Wood (1969). This function
is used in the Helmholtz integral for the inhomogeneous
medium to calculate the properties of the sound beam that
enters the bottom, is refracted in a circular arc, and
returns to the water column. The effects of roughness at
the interface are introduced using the analytical techniques
pioneered by Eckart (1953). The amplitude of the coherent
wave and the statistics of phase and amplitude fluctuations
will be discussed. Of particular interest are turbidite
layers since the acoustic velocity is less than that of
water and the normal reflection coefficient may be very
small.

This paper addresses the effects of bottom roughness on sound
which refracts in the ocean bottom. The analysis involved a number

of simplifying approximations which can be refined in later work.

Figure 1 displays the environmental parameters of concern to
the problem of rays that enter the bottom and are refracted back
into the water column. For numerical examples, values obtained by
Hamilton (1974) in the Japan Sea abyssal plain will be used. The
linearization of the square of the refractive index (Equation 1)

permits the solution to be expressed in terms of Airy functions, and

PIPES

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM
VELOCITY

1480 m/sec

DEPTH

SEDIMENT

GRADIENT
Tle steven

JAPAN SEA ABYSSAL PLAIN
HAMILTON (1974)

2
n? (2) = rich | =a = "ad (1)

v(O) + 5 av(0) (2)

V(Z)

lfa ~ 750m

Figure 1. ENVIRONMENTAL PARAMETERS

276

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

the parameter "a" can be related to the gradient in the sediment via
the approximation (Equation 2). Given a narrow beam impinging on
the bottom, the refracted path in the bottom will be as shown in
Figure 2 with subsequent multiple reflections. A phase shift of 1/2

occurs at each turning point in the sediment.

Two papers treat this problem. One is by Morris (1970) in which
reflection bottom-loss curves are computed for the linear gradiant
using the refracting layer plus an additional semi-infinite layer
below the sediment. In the second, Brekhovskikh (1960) treats the
case of a continuous velocity value across the interface (that is,
without the step discontinuity shown in Figure 1), Both papers treat
plane waves and obtain a complex reflection coefficient. Brekhovskikh
(Equation 3) assumes no losses in the bottom and, hence, the reflection
coefficient has unity magnitude. Morris (1970) adds attenuation in

the bottom, and the refracting waves have less than unity magnitude.

A major point of this paper is that if the problem is actually
for narrow beams, the result should be similar to a Rayleigh-type
plane-wave reflection coefficient, expandable in an infinite series
corresponding to the multiple bounces. This is analogous to the
treatment of a transmission line where the transmission loss through
it is calculated using a continuous wave but it can be expressed
as an infinite series of multiple reflections from the two ends
of the transmission line. When the result in Equation 3 is expanded
properly, it should become a reflection coefficient for the surface
with separate amplitudes for the successive waves corresponding to

the refracted and reflected paths in the bottom.

Brekhovskikh analyzes the case where there are no losses in the

bottom and the velocity is continuous from the water into the bottom.

207

THE EFFECT OF ROUGH INTERFACES ON SIGNALS

THAT PENETRATE THE BOTTOM

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278

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

He obtains the curve shown in Figure 3 for the dimensionless displace-
ment, u, of the wave that is refracted in the bottom as a function of
the dimensionless grazing angle, 8. A is the actual displacement, and
ao the grazing angle. For the model from Hamilton's paper, numerical
values are shown for the grazing angle in degrees and the horizontal

displacement in meters on the beam between entry and reemergence.

Brekhovskikh shows that the wave theory and the ray theory give
good agreement beyond 8 = 1. The subsequent discussion will be
restricted to grazing angles for which ray theory can be employed in

the bottom with some safety.

When there is attenuation in the bottom (Figure 4), there will be
losses on the refracted path and presumably the subsequent reflec-
tions will be of minor importance. In Morris's paper, the plane wave
reflection coefficient is used and the interference between the returned
paths after successive bounces is extremely sensitive to the grazing
angle. Hence, the resulting bottom-loss curves have a strong ripple
associated with the interference. If the interference is removed by
separating Paths 1 and 2, either in space or in time, (or if there is
an intromission condition with a very small reflection coefficient

for Path 1), then Path 2 should dominate the field.

The theory in which the velocity is strictly a linear function
of depth (rather than no linear as above) has been developed exten-
sively in a paper by Pekeris (1946) who solved the Green's function
for a point source in a linear gradient medium, and in a later paper
by D. H. Wood (1969). The Green's function is given by Equation (4)
in Figure 5 for a source at the origin of the coordinate system where

z is the depth and r is the distance from the source.

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

Ky = 21/WAVE LENGTH IN WATER
OS GRAZING ANGLE

A = DISPLACEMENT OF BEAM

1fa ~ 750m f = 1 kHz

Figure 3. BREKHOVSKIKH'sS COMPARISON FOR
WAVE THEORY AND RAY THEORY

280

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

D
<a oa RB
EEA a MAA
i SOLS
Soe
<=>
PATHS (1) AND (2) MAY BE SEPARATED IN TIME
OR IF
0-SEDIMENT VEL. WATER :
0-WATER VEL. SEDIMENT '

BOTTOM LOSS ON REFLECTION IS LARGE AND

PATH @) DOMINATES

Figure 4. CONDITIONS FOR THE REFRACTED PATH TO DOMINATE

281

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS

THAT PENETRATE THE BOTTOM

16
G( een ie el) Gna
tne 2, 2 yi nR
2
ms OF». = ee
On — ao al tanh R
v
oll(¢4)) =e Ae Lee
x? = x? af Sf ae 22
2
(e
R* = A + ye + (2+-2)

SOURCE AT (0,0,0)
YY = SOUND SPEED GRADIENT

SOURCES: PEKERIS, JASA 18, 295(1946)

D.H. WOOD, JASA 46, 1333(1969)

Figure 5. GREEN'S FUNCTION FOR A POINT SOURCE
IN A LINEAR GRADIENT MEDIUM

282

(4)

(5)

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

This particular form of the Green's function is not genuinely
symmetric in source and receiver as a Green's function should be.
The source is singled out by being placed at the origin leading to
the asymmetry in z which is the location of the field point. In
some subsequent expansions, the order of the z term will be switched
because of this asymmetry and this can be easily justified by
appealing to physical intuition. As is well known, in the linear
velocity medium rays are circles whose centers lie on the plane where
the velocity goes to zero. R, as seen in Figure 6, is the distance
from the "image" source a distance z above this plane to the observa-

tion point.

The view being taken of the bottom is shown in the middle portion
of Figure 6, where the ray enters in the first region, is refracted
downward, and emerges at the exit region through a different patch of
the ocean floor. It may also have reflected one or more times in the
middle region. A set of local coordinate systems is introduced in
the lower portion of Figure 6, where it is assumed that the ocean may
have a mean displacement in the entry region referred to the mean value
of the sea floor in the reflection region, and the exit region may
have yet another mean displacement. Hence, there will be phase
differences involved in the travel paths in the bottom associated
with these mean displacements. The phases can be given additional
statistical fluctuations associated with roughness in the local areas
where the sound enters the bottom, reflects, and emerges. The distance
L between the coordinate origins at the entry and exit regions is the

horizontal distance of the refracted beam in the bottom.

At this point, it is convenient to make a number of assumptions.
First, the regions should be well separated, that is, they should have

small linear dimensions compared to L. When L is large compared to

283

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

GE LINE OF CENTERS

LA = _- —_

2Y6

R
0
P(x,y,2Z)
Z
ENTRY REGION REFLECTION EXIT REGION
O MEAN LEVEL

Figure 6. GEOMETRY FOR APPLICATION OF GREEN'S FUNCTION

284

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

the size of the entry area, it will be shown that the Green's function
can be expanded about the origin, O,, and will be locally a plane wave

emerging at the surface.

The approximations used are listed in Figure 7. The increments
éx, dy, 6z, relate to the differences between the variable points in
the two little areas, and the z coordinate then contains the difference
in the local mean depths, SD. Expanding all quantities to linear
terms in 6x, dy, 6z, and 5p, the Green's function reduces to Equation
(6) in Figure 8, where the phase consists of two terms. a is the
phase length between 0, and O,, and o contains local departures

from a associated with entry and exit points (x), z1) and (x,, Zo)s
respectively. ® is quite accurately approximated by a local plane

wave (Equation 9) of emergent angle oo

Note that % is not symmetric in (x)7 z,) and (xo, Z5)- This is
the point alluded to earlier. If the field point (xo¢ Zz.) is taken
as a new source, then the behavior near the origin has the wrong
sign. To remedy this, the first term is always the field point

and the second term is always the source coordinate.

This result is summarized in Figure 9. A source ray enters the
bottom at point Q at some angle Oe emerging at point P at the same
angle. The variable phase delays associated with roughness at points

Q and P can then be added to the geometric phase delay via -

The Helmholtz formula (Equation 11) in Figure 10 is used to
calculate the field at the point P integrated over the area of
insonification. For the Green's function, the linear approximation
is used which simplifies the normal gradient in the integral leading

to Equation 12.

285

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

eee) oo (Gy) > en SD oe

(@
R2= (L + Sx) 2 ar (sy)? (<2 + 6D + sz)?

DEFINE ee
Y

BASIC APPROXIMATIONS

I 2S Oeisp Ch7n Op OD)

(e Cc
Dri) aS 22*30) A ee)
2y 2y ii
2 -1
W ~~ @ y ~ 1 sec
ote =
WwW > 600 sec

AND EXPANDING: tanh 2

wih

TN, O35) eevee ear

Figure 7. APPROXIMATIONS

286

TO LINEAR TERMS

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

G= 70 id _+i¢6,'
yR_L fe) 1 (6)
e)
? Sk sok | te een, Viet pi, Vo 10D
YR aR yR_2 R if (7)
[e) [e) (e)

fe 2, 2
= Lge Om ike L_ (:, a (8)

FOR THE PHYSICAL PARAMETERS IN THE PROBLEM, THE
LATTER CAN BE APPROXIMATED VERY CLOSELY BY

%) = Ky | sin Bs f:,-*,) -cos eh (22-21) | (9)

Figure 8. GREEN'S FUNCTION WITH APPROXIMATIONS

287

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

eal ik, | i 8, (x-»,)- cos 8, (-.-m) (10)

BEHAVES PROPERLY AT P
BUT IF WE THINK OF P AS A SOURCE AND Q AS THE DETECTOR,
VAs “ealiavels ¥4

2 1 MUST BE INTERCHANGED. THIS IS IMPORTANT IN
THE HELMHOLTZ FORMULATION.

Figure 9. CONVENTION REGARDING SOURCE AND FIELD POINTS
TO REMEDY ASYMMETRY IN GREEN'S FUNCTION

288

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

WATER
SEDIMENT
his z ra

THE HELMHOLTZ INTEGRAL FOR POINTS P AND Q BELOW ed Is

o(P) = an | op ey Se re (11)

NOW Oye — oO] and
on dZ4
+ Io c= + ik cos @_G,
an OZ O e
1
SO
ea cK)
o(P) = aT i 1 32, + i ko cos 68, $ cae (12)

IF ¢ IS LOCALLY A PLANE WAVE WITH ANGLE OF
INCIDENCE, Cees IT WILL BE REFRACTED INTO THE BOTTOM
WITH REFRACTION ANGLE Op:

Figure 10. HELMHOLTZ AND GREEN'S FUNCTIONS

289

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

The variable displacement of the surface in Figure 1l, at the
point of entry, produces a variable phase delay 6 (Equation 13).
To the simplest approximation it is actually a difference of the
slant path in the water associated with the entry angle and the
slant path in the sediment associated with the refracted angle.
This makes the effect much smaller than in scattering, say, froma
free surface or from a reflecting bottom. That is, only the differ-
ence in the acoustic delays in the two media accumulates, so that a
large surface displacement actually produces a relatively small
change in the phase. Hence, Equation (13) is the variable phase
to be inserted across the area of integration, being the random
displacement of the surface. Again assuming that the normal gradient
of the field in the bottom is the vertical gradient, there are two
final approximations: first, that the angle oe in the Green's
function is the same as the refracted angle o. of the wave entering
the bottom; and second, that the wavenumber k, for the refracted
wave and Ko for the Green's function are the same. With this
approximation the field (P) is expressed in Equation (14) as the
integral over the insonified region of the refracted wave incident
on the bottom times the Green's function integrated over the insoni-

fied area.

Hence, the wave impinging on the bottom is refracted in the
bottom yielding >. There is a phase variation with xX across the area
of insonification, but the Green's function to the linear approxima-
tion used here has exactly the same phase variation because of the
agreement of phase at the boundary. That is, the X variation of phase
in the one function is exactly canceled by the variation in the
Green's function, leaving only the variable phase delay associated

with roughness.

290

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

A VERTICAL DISPLACEMENT C(x), Y,) WILL PRODUCE
A PHASE DELAY, 6
211 iL 2II al
=f ee > G (13)
‘ cos OF AB cos Oo,
WHERE W: WATER B: BOTTOM

FURTHER, IN THE MEDIUM AT THE ANGLE On:

aa elk kp cos 026

WITHIN THE ACCURACY OF THE LINEAR APPROXIMATION,

Oe i eB, re 7 be
SO
ma ak ei 2.0COs 10
g(P) = —#=,,—— | ocae (14)

Figure 11. VARIABLE PHASE DELAY AND ITS EFFECT ON $¢

29)

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

The integral is performed in Figure 12 where m designates the
phase delay per unit displacement. By introducing plane-wave
approximations, the field » at the point P just under the emergent
area is given by the integral shown in Equation (15) over the area

of insonification.

The last factor in Equation (15) represents the local plane

wave about the field point P, emerging at the exit region.

The integral is a stochastic integral and, if the insonified
area is large compared to the correlation distance of the displace-
ment, C, the exponential can be expanded in a convergent series
(Equation 16). <> is the average value of the displacement and

since the local origin is on the mean surface, <> = 0.

Hence, there is no phase shift associated with entry into the
bottom. <t2> is the mean square displacement and results in a loss

of amplitude. For abyssal plains (<z2>)4 is of the order of 3 to
10 centimeters and there is very small loss of amplitude associated
with entry into the bottom. Hence, there is a coherent wave

arriving at the exit region with very little loss.

The same type of analysis can be repeated almost word for word
for the emergent ray, resulting in a second slight loss of amplitude
associated with the mean square displacement at exit region.
Typically, the entry and exit regions are far enough apart (several
hundred meters) that there is no statistical correlation between

<t*> at point Q.

In summary, it appears that moderate roughness at the bottom-
water interface will produce essentially no loss of amplitude on

entering or leaving the bottom, and the strength of the refracted

292

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

LET

TRANSMISSION COEFFICIENT
a = AREA OF INSONIFICATION

211 1 . al
rA.. cos 8 NETCOSaG
WwW WwW B

THE FIELD AT P IS THE PLANE WAVE

: ik_|x,sin8§_-z.,cos8@
ime (x,, y,) | 2 B “2 |

o(P)= C 2
= e ax, dy, e
(15)
a
WHE RE
G = 2 ale Ss kp cos on Ra 10)
YR, 21
NOW
img i 2 (
e dx, dy, = aJl1 - im<t>-sm*<z7?>+ <--> (16)
al 2 j
es <T> =o NO PHASE SHIFT
1f2
<t?> ~ 0.03=0.1 m, ABYSSAL PLAINS

Figure 12. RESULTING FORMS

293

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

path will depend on the reflection coefficient at the interface and

the attenuation the path receives in the sediment.

REFERENCES

Brekhovskikh, L. M., Waves in Layered Media, pp. 117-122, 189-193,
Academic Press, 1960.

Eekane, Cs, Wie Acoust. Soc. Am." 25,) 560, L953"

Hamilton, E. L., "Geoacoustic Models of the Sea Floor," in Physics of
Sound in Marine Sediments, Ed. Loyd Hampton, Plenum Press, 1974.

Morris, H., J. Acoustic Soc. Am. 48, 1198, 1970.
Pekeris, C. L., J. Acoust. Soc. Am. 18, 295, 1946.

Urick, R. J., “Underwater Sound Transmission Through the Ocean Floor,"

in Physics of Sound in Marine Sediments, Ed. Loyd Hampton, Plenum
Press, 1974.

Wood, D. H., J. Acoust. Soc. Am. 46, 1333, 1969.

294

HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS
THAT PENETRATE THE BOTTOM

DISCUSSION

DR. IRA DYER (Department of Ocean Engineering, Massachusetts
Institute of Technology): I understand your conclusion to be that
moderate roughness does not substantially affect the computations of

refracted paths in the bottom sediments.

DR. C. W. HORTON, SR.: Yes.

DR. DYER: Does this include the effect of scattering of this
energy outside the angles at which you might expect to receive these

bottom refracted paths?

DR. HORTON: I should have mentioned in my discussion and didn't
that this refraction path is essentially unique. That is to say, for
one configuration of source and receiver in the water there will be
only one path through the bottom that gives you the travel time that

you will see. This is borne out by the experimental data.

The loss of amplitude I referred to represents, I believe, all
the data that is scattered in directions other than the refracted path.
So they are essentially refracted out of this acoustic bundle and don't

arrive at the emerging point.
DR. DYER: And nonetheless small?

DR. HORTON: Nonetheless small for the moderate amplitudes.

295

BOTTOM PROPERTIES FOR
LONG-RANGE PROPAGATION PREDICTION

Aubrey L. Anderson

Applied Research Laboratories
The University of Texas at Austin
Austin, Texas

Loss of acoustical energy propagating to long ranges in the
ocean is predicted with computerized mathematical models
(propagation models), many of which treat the ocean bottom
as either a reflecting interface or as part of the propaga-
tion path (a penetrable boundary). In many models, the
bottom is included as an interface which is characterized
by a plane-wave amplitude reflection coefficient. The
reflection coefficient (or bottom loss) is obtained either
from bottom-loss measurements or from calculations using
mathematical models of the bottom as an acoustical reflec-
tor (bottom loss models). Bottom-loss models require, as
input, detailed information on the physical properties and
layering of the bottom material. This presentation relates
the topics of bottom-loss measurements and models, bottom
physical properties and topography to long-range propaga-
tion. Sensitivity of propagation loss to bottom parameters
is discussed.

INTRODUCTION

The ocean bottom is one of the boundaries with which a propagating
underwater sound wave may interact. In some cases our present tech-
niques for including bottom interaction in propagation models do not
allow accurate prediction of propagation when bottom interaction is
significant. Perhaps this is because our input information is in-
complete, or perhaps our method of including the bottom influence

should be refined.

Several topics are of interest in the bottom interaction problem.

These include:

2917

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

@ Models of bottom interaction presently used in propaga-
tion models

e Models that allow the inclusion of bottom effects: bottom-
loss models and geoacoustic models.

e The order of magnitude of observed and predicted bottom
loss values

e Measurement techniques and presently available data for
acoustical and other physical properties of the bottom

e The sensitivity of bottom loss to variations of the input
sediment parameters

e The sensitivity of propagation predictions to variations
in the bottom information.

Unresolved issues concerning bottom properties for long-range

propagation include the following questions:

@ What is the sensitivity of predicted long-range, low-
frequency propagation loss to variations in the sediment,
either bottom loss or physical parameters?

e What is the sensitivity of bottom loss to sediment param-
eter variation?

e To what depth and in what detail do we need sediment
information to predict bottom loss?

@ What information do we now have and what techniques need
further measurement?

Two general techniques treat mathematical propagation problems.
These are, of course, ray theory and wave theory. Each technique may
treat the bottom as a reflecting surface or as part of the propagation

path.

For example, in the ray-theory models we identify eigenrays by

searching through ray families until we find two that bracket a

298

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

receiving point and then converge on an eigenray. Once the eigenrays
are identified, one way of treating the influence of the boundaries
and in particular the bottom is through an interface reflection co-

efficient, a Rayleigh plane-wave reflection coefficient.

Either method of treating the bottom requires more detailed in-
formation about the physical properties of the bottom sediments. These
physical properties include acoustical properties such as speed of
propagation and attenuation and are combined in what Hamilton (1974)

calls a geoacoustic model.

In some cases when the boundary is treated as a reflecting inter-
face, we can go through an intermediate model, feeding the geoacoustic
model information into a mathematical model for computing bottom loss.
An alternative is to structure the measurements of bottom loss into an

empirical model.

BOTTOM-LOSS MODELS

Figure 1 illustrates some of the bottom-loss models. Standard
empirical bottom-loss models consist of tables of bottom loss versus
grazing angle. Probably the earliest of these came from the AMOS
program, another set was developed at Fleet Numerical Weather Central
based on the MGS data, and some have been based on the FASOR data.
NAVOCEANO also has a set. Other measurement programs have produced
what can be considered as empirical bottom-loss models at various

frequencies.

Mathematical models progress through a series of increasing
complexity using plane interfaces, plane layers, plane waves. Models
with liquid layers can progress to layered models that support shear

waves. More complex models may have gradients of the acoustical

299)

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

A. MEASUREMENTS (EMPIRICAL MODELS)

1. "STANDARD" TABLES AMOS
MGS
FASOR

2. SPECTALIZED

B. MATHEMATICAL MODELS
1. PLANE INTERFACE
2. ROUGH INTERFACE
3. PENETRABLE ROUGH INTERFACE
4. SEDIMENT PARAMETERS» GEOACOUSTIC MODEL

Figure 1. BOTTOM-LOSS MODELS

300

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

parameters with depth. Effects of rough interfaces have been studied

by Horton (1976). All of these require some type of geoacoustic model.

GEOACOUSTIC MODEL

A geoacoustic model may be described as a quantitative descrip-
tion of the pertinent sediment and water parameters, particularly the

former. This description includes at least the following:

e Layering that exists and depths of these layers
e Compressional-wave speed and attentuation

e Shear-wave speed and attenuation

@® Density

e Gradients, if they exist, of speed and density

@ Bottom topography.

BOTTOM- LOSS VALUES

Figure 2, an example of an empirical model, shows the low-
frequency bottom loss versus grazing angle model. There are several
notable features of these curves. For the lower three curves, the
bottom loss goes to zero between 10 and 20 degrees. This feature
indicates a critical angle effect which implies no attenuation in the
sediments. But when you put attenuation in, you don't see this zero
bottom loss except at zero grazing angle. For the two higher bottom
loss curves, we see that the loss does not go to zero even at zero
grazing angle and this implies considerable influence of topography

in these two classes of the empirical model.

Figure 3 is a mean bottom-loss curve (Urick, 1974). Observe

some differences from numbers in Figure 2.

301

ANDERSON:

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

)

BOTTOM-LOSS CURVES

50
(deg.

20 30 40
GRAZING ANGLE

al}

5 e °
(dP) SSOT WOLLOG

302

Sei al

Figure 2.

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

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303

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

Figure 4 presents a comparison (Bucker, 1974) with data anda
computation with a linear-gradient model of Morris. The data are for
one-third octave at 50 Hz and the theory (solid line) is for con-

tinuous wave.

BOTTOM PHYSICAL PARAMETERS

There are essentially three depth intervals into which sediment
can be broken for measurement techniques: surficial sediments, corer-

depth sediments, and deeper sediments.

In the surficial sediments, acoustical properties have been
measured fairly extensively, especially the compressional-wave speed
at higher frequencies. The surficial sediment includes about the
first meter. These have been probed by everything from divers to the
diving saucer, with various types of probes, and with self-operating
units. Considerable information is available in the literature about
some of the parameters. Physical properties have also been studied

extensively using grab samples and cores.

Corer-depth sediments extend from the one-meter depth to perhaps
30 or 40 meters for the very long cores. Acoustical information is
available to this depth from high-resolution sub-bottom profiling in
some regions. Core sediment samples are measured in the laboratory
and these values are then extrapolated to in situ values. Compres-
sional wave speed and attenuation are studied, especially again at

high frequencies.

Recently, a different technique was instituted for corer-depth
sediments by ARL/Austin. It is a device called a profilometer which
projects a pulse across the diameter of the corer as the core is be-
ing taken and measures compressional-wave speed and attenuation in

the sediment at a carrier frequency of 200 kHz.

304

ANDERSON:

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

oO
m

GRAZING ANGLE (DEG)
BOTTOM-LOSS VS GRAZING ANGLE

co)
4
3
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3
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jaa)
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NOILO31434 WOLLOS 8

3105

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

Measurement information on the deeper sediments comes from sub-
bottom profiling of one kind or another; e.g., reflection and refrac-
tion profiling. Most of the information concerns the speed of
propagation of compressional waves which is converted into information

about depths to sub-bottom reflectors.

Physical information comes from drilled samples. Considerable
information is being compiled by the Deep Sea Drilling Project. An

interesting idea might be velocity-logging these holes.

Most available data on the bottom physical and acoustical param-

eter values are for high frequencies and compressional-wave speed.

In surface sediments, Hamilton (1974) has added to our knowledge
of in situ values of surficial sediment speed and of techniques for

extrapolating laboratory measurements to in situ values.

Figure 5 represents something like 3,000 measurements of speed
of propagation of compressional waves, and they are plotted as a ratio
of speed of propagation in the sediment to that in the water. They
show the well known, somewhat well defined relationship between speed
of compressional-wave propagation in sediments and porosity. These

are high-frequency values.

The values go from something like 0.95 or about 5 percent lower
than the value in bottom water to almost 30 percent higher than the
value in bottom water. Sound-speed values outside this range of
variation are anomalous for unconsolidated sediments. Such values are
usually associated with gas in the sediment in some form. In shallow
water sediments, gas will exist as a phase, a gaseous phase, and it
will decrease the value of speed of propagation. In deeper water, it

is more likely to exist as a gas hydrate or clathrate, and it will

306

RELATIVE SOUND SPEED - C,/C,,

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

C, = SEDIMENT SOUND SPEED
Cy, = WATER SOUND SPEED

—_
_—

ro)

0.9
20 40

80 100
POROSITY — PERCENT

Figure 5. RELATIVE SOUND SPEED (RATIO OF SPEED IN
SEDIMENT TO SPEED IN WATER) VS SEDIMENT
POROSITY (AFTER AKAL, 1972)

307

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

greatly increase the speed of propagation. In the latter case, the

sediment essentially looks like it is partially lithified.

Figure 6 shows the interval velocity for a region and Hamilton's
estimate of the best fit line to the instantaneous velocity data. In
this example, the instantaneous velocity, which is the one that would
go into a propagation model, is nonlinear over an extended depth
interval. However, the interval velocity, which is the average

velocity over the measured depth interval, remains somewhat linear.

Compressional-wave attenuation is another important parameter.
Figure 7 is a compilation from a large number of sources of data for
acoustical attenuation in dB per meter versus frequency. These re-
sults are for measurements which were made in clays and silts. It

is a presentation which is similar to what Hamilton uses.

Several things can be seen. One is the order of magnitude of
the attenuation. Another is the absence of any data for anything

below 1 kHz.

Another observation is that over short frequency intervals in
any given sediment the attenuation may not vary linearly with fre-
quency. But if we take the overall behavior as we go down the graph,
attenuation varies linearly with frequency. If this is true, and
certainly these data seem to indicate that it is, then it suggests
a way to get a number for the attenuation at low frequencies. We
must decide what value we are going to accept for attenuation at
some high frequency and extrapolate linearly downward to a lower
frequency of interest. We hope to improve upon this extrapolation

in the future.

If we accept that attenuation is described as a linear function

of frequency — that is, the attenuation coefficient is equal to

308

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

0) T l = if i
0.05
A ° : |
e @ e
0.10 se Nae 4
e t )
_\ 8 eo’,
V V e°
0.15 a e e 4
NORTH °
5 (SHALLOW) o*\e
2 0:20 as A =|
| t )
= O25 = e
- vs e e ® 7
= &
= 0.30 } o\ 8 4
= ”
= 0.35 ® as 4
S
Ww
Pa _
6 0.40 e
. Vv
0.45 - ol
e
0.50 |
CENTRAL
0.55
0.60 ie il has a | es “al
1:4 1.6 1.8 2.0 22 2.4

VELOCITY — km/sec
Figure 6. TRAVEL TIME VS VELOCITY

(Source: Hamilton, 1974)

309

ATTENUATION - dB/m

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

= lie a al le ll ae [= ieeliey
e KAOLINITE, URICK, 1948

BOUNDS OF SHUMWAY
DATA, 1960

EMSWORTH MUD, WOOD AND WESTON, 1964
KAOLINITE, HAMPTON, 1967 4

CLAYEY SILT, NORTH ATLANTIC AND
MEDITERRANEAN, BENNET, 1967

CLAYEY SILT, NORTH AND BALTIC SEAS 4
(AVERAGE), ULONSKA, 1968 al

SILTY CLAY }

100

CLAYEY SILT
STIFF CLAYEY SILT

NORTH ATLANTIC, McCANN Va
AND McCANN, 1969

4
McLEROY AND
DeLEACH, 1968

-dPD 4 xX

(oXe)

0) | a | es ! Le La = 1 SEES
1 10 100 1000

FREQUENCY - kHz
Figure 7. ACOUSTICAL ATTENUATION VS FREQUENCY IN
CLAYS AND SILTS (LESS THAN 1% SAND)

310

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

some K' times frequency to the first power — and we examine data that
Hamilton (1974) has presented on attenuation for Pacific sediments and
data that Smith has presented for attenuation in Atlantic sediments,
then we find that we can plot the values of the coefficient K' versus
the mean grain size of the sediment, Figure 8. The resulting rela-
tionship will help us select the K' to be extrapolated as a linear

function of frequency.

Other bottom parameters in the geoacoustic model include bulk
wet density, which is usually measured with samples, and shear-wave
speed which has been measured only in a very limited manner. Bucker
(1974) appears to be one of the few who has actually made these
measurements. He measured Stonely waves and interpreted them in terms

of velocity of propagation of shear waves.

The answer to the question raised earlier about whether these
sediments behave as liquids or as solids depends on what you mean by
the question. If the question is "Do shear waves propagate?" the
answer depends on whether there is a finite value of dynamic shear
modulus. Values of dynamic shear modulus have been measured in most
ocean sediments somewhere on the order of 10° to 107 dynes per square
centimeter. Propagation speeds of the shear waves are something on
the order of a tenth of the value of propagation speeds for the longi-

tudinal waves.

In near-shore sediments, very high porosity sediments, harbors
and lagoons, we find even lower values of shear modulus. The lowest
values of dynamic shear modulus are exhibited by freshly mixed, pure
laboratory clays like Kaolinite, for which values of less than 10
meters per second are predicted for shear-wave speeds from measured

values of dynamic shear modulus.

311

ANDERSON:

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

O-——O E.L. HAMILTON
+--+ D.T. SMITH
@ = K’f dB/m

5 6 7 8

0 1 2 3 4
MEAN GRAIN SIZE -
Figure 8. MEAN GRAIN SIZE VS ATTENUATION CONSTANT K'!

312

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

Shear wave attenuation measurements are few. Some measurements
of the complex dynamic shear modulus allow prediction of a shear wave

attenuation.

What does all this have to do with bottom loss and with propaga-
tion loss at low frequencies? What is the sensitivity of this thing
we call bottom loss (which is an input to ray-theory models) to

variations in sediment parameters?

The simplest reflection models, using a liquid layer without any
attenuation, a single layer overlain by water, can fit some of the
things that we see in Figure 9. Judicious selection of the sound-
speed ratio can make the critical angles fit, and juggling the density
ratio can cause the bottom-loss values at normal incidence to fit.
Unfortunately, when this is done, the grazing-angle segment just above
the critical angle does not fit these data. This seems to indicate
that the single bottom layer is far too simple a model. Disagreement
is not as bad as one might expect. The important thing is that this
shows realistic values of speed of propagation and of density for
bottom sediments. One problem, however, is that some of the bottom
loss obviously is going to be contributed by topographic effects which

are not included here.

Figure 10 shows the results for a water layer overlying a two-
layer bottom. This three-layer model, with a clay overlying sand in
the bottom, is shown merely to indicate the type of variation that
is shown at 100 Hz for a value of attenuation obtained by the extrapo-
lation process mentioned earlier. The sound speeds are 1,501 meters
per second in the water, 1,531 meters per second in the clay, and
1,657 in the silty sand, with realistic values for density and with

a 100-meter thickness for the clay layer.

313

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

“06

“08

WOLLOd YAAVTI GINOIT ATONIS
ATONY ONIZVWYD SNSYHA SSO'T-WOLLOd

("bep) AIONY ONIZWYD

OVE Om OS 0°OS O°OV O°OE

"6 oanbtgy

SSOT WOLLOd

(dP)

314

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

(WOOT=SSHNMOIHL YHAVTI AWIO)
THCOW YSAVI FSNHL
AFIONY ONIZWUD SA SSOI-WOLLOG

"OT eanbta

(°bep) AFIONY ONIZVYD

OF Oz OmOg OROS 0°OP

O°OE 0°02

SSOT WOLLOd

315

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

We see the indicated interference structure in the variation of
bottom loss with grazing angle. Why do this? If we extend the depth
of the layer, we will see to what depth we must go before we begin to

lose the effect of the clay/sand interface.

Figure 11 shows the results when the intermediate layer is 200
meters thick. The interference structure is reduced, but still
present. Figure 12 shows results for a layer thickness of 500 meters.
Figure 13 is for 1,000 meters. The interference structure is gone.
Thus, for a layer of this thickness and the assumed attenuation, the
Rayleigh reflection-coefficient model indicates that the lower interface

with a sand layer does not influence the bottom loss.

Doing this for the same type of clay overlying basalt, where we
have a considerable impedance contrast between the clay and the basalt,

the following results are calculated.

In Figure 14, the highly variable curve is for 70 meters of clay
overlying basalt. The smooth curve is for a 1,000-meter thick clay

layer over basalt.

These results indicate that if the reflection model used here
were valid for bottom regions described by the parameters assumed
here for the clay layer, and if we knew the information about the
sediment column to 1,000 meters depth, we wouldn't have to know

anything about it from there on down.

Also, we have seen instances where there is considerable energy
return from 2 to 3 kilometers. The result described above is critic-—-
ally dependent on the value of attenuation that is used for the layer.
Also, the model used for the calculations does not include gradients
in the layers. This work is being extended to include gradients in

the bottom.

316

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

(WOOZT=SSHENMOIHL YRAVT AVIO)
THGOW UAAVI ATNHL

FIONY ONIZWYD SA SSOTI-WOLLOG

(*bap) AZIONY DSNIZWEO

O°OL 0°09 OOS 0°O

"TT eanbtg

O°O€ OF 0G

SSOT WOLLOd

(dp)

Silky

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

(WO0S =SSENMOIHL

UAAVT AVTIO)

THGOW YHAVT Sed
SSOI-WOLLOG ‘ZI eanbty

ATONVY ONIZVWUS SA

(*6ep)
O°OL 0°09 0° 0S

ATONY ONIZVYO
0° Ov Om0le

OF 0S

OmOG

ORO

SSOT WOLLOd

(dp)

318

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

(*WwOOOT=HLddd WHAWT AVTIO)
THGOW USAVI ASHHL

ANTONY ONIZWYD SA SSOT-WOLLOG “ET eanbta
(*bep) AIONY ONIZVYD
0°09 0°0S 0°OP O70 O° O02

SSOT WOLLOd

(dP)

BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

ANDERSON:

LIVSVd ONIAVTIGHAO AVIO
*HTIONY ONIZVWYD SA SSOT-WOLLOG

(°bep) ATONY SNIZVWAD
0°09 0°0S 0°OP O°O€

“pT eanbty

SSOT WOLLOd

(dP)

320

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

REF ERENCES

Bucker, H. P., "Sound Propagation Calculations Using Bottom Reflection
Functions," in Physics of Sound in Marine Sediments, L. Hampton (ed.),
Plenum Press, 1974.

Hamilton, E. L., "Geoacoustic Models of the Sea Floor," in Physics
of Sound in Marine Sediments, L. Hampton (ed.), Plenum Press, 1974.

Horton, C. W., These Proceedings.

Urick, R. J., "Underwater Sound Transmission Through the Ocean Floor,"
in Physics of Sound in Marine Sediments, L. Hampton (ed.), Plenum
Press, 1974.

DISCUSSION

Mr. Charles Spofford (Office of Naval Research): What value do

you get by extrapolating Figure 7 to 100 Hz?

Dr. A. L. Anderson: About 0.026 dB per meter, something like
that.

Mr. Spofford: We have seen data in certain areas where there is
a very thick unconsolidated sediment. Assuming a 20-degree ray and a
refracting gradient of one in the bottom, the ray spends about 1,000

meters in the bottom per bounce.

We have seen data where essentially that ray appears to have
bounced up to about 10 or 20 times even out to 200, 250, and 300 miles
without suffering appreciable loss. Figure 7 would lead to about
20 dB in 200 miles. I would say if it has lost anything it might be
about 2 dB. It is that little.

321

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

Dr. A. L. Anderson: I think there are two significant points.
One is, as I pointed out, this calculation was made for a model with-
out gradients, so we need to remember that, although it is not
particularly germane to your point, Also, I think that you may very
well have a good, if not the only, way of actually measuring attenu-

ation at a given frequency in the sediments.

Dr. D. C. Stickler (Applied Research Laboratory, Pennsylvania
State University): I would like to point out that in your models of
plane-wave reflection coefficients, your layer media, that some of

those same effects can be observed even without the layering.

If you consider the full effect of a point source in the iso-
velocity halfspace and higher speed bottom, you can observe some of
these oscillations away from the grazing angle and the breakaway from

the O dB loss above the line.

Dr. A. L. Anderson: Precisely, which says you must consider

something other than a plane-wave reflection coefficient.

Dr. D. C. Stickler: Yes. If you do the full-wave solution for
a point source in isovelocity halfspace over a higher speed iso-
velocity halfspace and examine just the reflected field, then these
oscillations above grazing are present and the breakaway from the
zero reflection coefficient is also observed and is not related to

layering at all and is also frequency-dependent.

Dr. W. H. Munk (Institute of Geophysics and Planetary Physics,
University of California at San Diego): Question based on ignorance.
Are there good statistical models of the sea bottom? And, I mean it
in the sense of existing statistical models of the sea surface that
I am familiar with which have indicated that scattering from an

angle of incidence steeper than the root mean square slope behaves

322

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

entirely differently than scattering at angles of incidence much less
than the root mean square slope. One region is specular and in
another case it is backscattering. Are there similar kinds of con-

siderations for sea-floor scattering?

Dr. M. Schulkin (Naval Oceanographic Office): Not quite, but
there is a spectrum of the sea floor bottom that has been proposed —

Reo

Dr. Munk: Is that taken seriously?

Dr. Schulkin: Well, until there is something to replace it,

it is semi-serious.

Dr. Munk: It goes down to what short wave length? Two hundred

meters?
Dr. Schulkin: Yes.

Dr. Donald Ross (Tetra Tech, Inc.): May I make a comment on
some model work? We are closely associated with the work that is go-
ing on at Naval Undersea Center in which a computer model for propa-
gation, FACT model and FACT extended, is being compared with hundreds
of experimental measurements in the low-frequency regions and we are
finding that the models do well in the region in which you have
refracted rays and that they are extremely sensitive to bottom loss
in the region in which the bottom is involved, that the bottom loss
is apparently averaging of the order of 1 dB, and that a quarter of
a GB difference in the loss per bottom bounce makes a significant
difference in the results that you get when you are comparing the

experiment and the propagation model.

323

ANDERSON: BOTTOM PROPERTIES FOR LONG-RANGE PROPAGATION PREDICTION

In all of your graphs it is very hard to see a quarter of a

dB and in order to make significant calculations, we need bottom loss

to a quarter dB.

I think that this may mean that the way to get it is to make
measurements of propagation and deduce backwards what the bottom
loss must have been rather than to try to calculate it or make direct

measurements of bottom loss.

324

FORWARD SCATTERED LOW-FREQUENCY SOUND
FROM THE SEA SURFACE

W. I. Roderick

Naval Underwater Systems Center
New London Laboratory
New London, Connecticut

Low-frequency propagation over long ranges can have propa-
gation paths that interact with the time-varying sea
surface. Theoretical predictions and experimental obser-
vations of specularly reflected CW acoustic signals indicate
that the long gravity waves on the sea surface modulate the
amplitude and phase of the incident signal. The Doppler
spectrum of the modulated signal consists of a discrete
frequency component centered at the carrier and a continuous
spectrum that is positioned symmetrically about the carrier.
The continuous spectrum consists of energy that has been
scattered close to the specular direction and that, when
summed with the specularly reflected signal produces
amplitude and phase modulation. A review is given of
important contributions to our understanding of the forward
scattered Doppler spectrum and its functional relationship
to geometrical, acoustical, and sea surface parameters.

This paper is an informal review of one particular aspect of
forward scattered sound from the sea surface and that is the Doppler
spectrum that would be received in the specular direction. The
Doppler spectrum is the spectrum resulting from amplitude and phase
modulation of an acoustic signal reflected and scattered from a time

varying surface.

About 1965, Wysor Marsh looked at two separate aspects of
scattering, one of which was the Doppler spectrum (Marsh and Kuo,
1965). It is interesting to look at that report written 9 years
ago and at a time when there had been no prior direct measurement of

the Doppler spectrum in the specular direction. Wysor observed in

325

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

some low-frequency, long-range propagation data, published by Ken
Mackenzie (1962), that the envelope of the signals had periods on the
order of ocean swell waves and the spectrum looked narrowband. Based
on a resonance scattering theory he developed in the early 60s, he
derived the Dopler spectrum which was good to first order effects and
included multiple-bounce surface interactions. Remarkable intuitive
reasoning at that time put the theoretical prediction of the Doppler

spectrum ahead of the experimental evidence.

What criterion do we use to distinguish between a rough and smooth
surface? Lord Rayleigh took a simple approach (Beckmann and Spizzi-
chino, 1963), pictured in Figure 1. He simply considered the phase
difference between rays reflected from an uneven surface. For the
wave shown, the crest to trough height is h, the grazing of the
acoustic rays is $¢, and the acoustic wave length is A — Walter Munk
might say here that this does not look like a sea surface wave, but
he must consider that the wave was measured on the east coast of the

United States.

Very simply, Rayleigh reasoned that if the phase difference is
near zero, then the surface isn't very rough, that is, the path-
length difference is small. As the phase difference approaches T
there will be cancellation of energy in that direction and hence the
energy must have been scattered elsewhere — this would constitute a
rough surface. A criterion to separate smooth and rough surfaces is
to choose a point midway between zero and 7, say T/2. As you can see,
the wave length, grazing angle, and wave height must be specified to
define the roughness. These three parameters crop up again with the
same relationship in more elaborate scattering models. As an example
of the above, consider long-range propagation with a 5-degree grazing

angle at the surface and an rms wave height of 2 feet — this

326

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

AOVIUNS NHARSNN CHZITVACI NV WOdd DNIXALLVOS
Ly=0
( puisq) Mi

‘TT oanbty

327

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

condition would require an acoustic frequency of less than 1,000 Hz to

constitute a low-roughness surface.

About 1966, Allen Ellinthorpe published an article (Ellinthorpe,
1966) on sea-surface induced frequency smear. It should be noted that
Doppler spectrum, frequency smear, and frequency spread all have the
same meaning. Ellinthorpe was interested in determining the integra-
tion time for a communication system and performed a surface scattering
experiment in Bermuda. To compare the experimental results, he derived
the Doppler spectrum of the forward scattered sound based on a phase
modulation technique. Assumptions were made that the surface scattered
signal was only phase modulated, the surface wave height h was a
Gaussian random variable, and the power spectral density of the surface
waves was given by a Bretschneider spectrum. With these assumptions,
he uses an equation derived by Middleton to determine the power
spectral density of a signal that is phase modulated by a random vari-
able that has a known power spectral density. The phase modulation

index is given by a.

Ellinthorpe compares the theoretical results that were derived,
based on the Middleton equation, to experimental data measured off
the coast of Bermuda. In Figure 2, I have selected a comparison
made at two frequencies. You can see that the agreement is close
in the spectral peaks, but the spectral width of the predicted is
narrower than the measured. In general, this is true of all his
predictions. The predictions do not give an absolute value of the
energy in the carrier and sidebands, and the predictions were obtained
by varying the parameters to obtain a best fit. No oceanographic data
were available. There is something of interest that will come up
later — the sidebands of the measured spectrum are asymmetrical as

shown for the carrier frequency of 856 Hz.

328

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

200 100 0 100 =. 200
RELATIVE FREQUENCY IN MHz

NORMALIZED SPECTRUM LEVEL IN dB

=32

“40°00 400.0400 800

RELATIVE FREQUENCY IN MHz

Figure 2. DOPPLER SPECTRUM AT TWO FREQUENCIES
(from Ellinthorpe, 1966)

829

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

In 1967 B. E. Parkins of BTL published an article (Parkins, 1967)
on the Doppler spectrum of scattered sound from a slightly rough and a
very rough sea surface. He used what is termed a physical optics
approach in his derivation, which is based on the Helmholtz Integral.
The approach is based on some fundamental concepts introduced by Carl
Eckart in 1953 on scattering from the sea surface (Eckart, 1953).
Eckart's approach is one of the most elegant treatments you will find
on surface scattering and before discussing Parkins derivation, we will

review some of these concepts.

The evaluation of the Helmholtz Integral

oP, exp (ikr)) 9 exp (ikr))
oie eg iece|| roe pal diame oermemeas foo

requires knowledge of two boundary conditions — the value of the re-
radiated pressure PL on the surface and the value of the derivative of
the reradiated pressure dP, /9v on the surface with respect to the
surface normal. By assuming that the sea surface is pressure release,
the reradiated pressure is set equal to the incident pressure with a

180-degree phase shift.

P +P = Oons (2)
fe) all

To find the value of the derivative of the reradiated pressure with
respect to the normal to the surface, Eckart assumes that the slope
of the surface irregularities is small and finds the derivative with
respect to the normal to the plane surface on which the gentle undu-

lations are superimposed.

— =| SS on S (3)

33:0

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

There are certain inherent assumptions with these boundary condi-
tions and further approximations were generally made to obtain tractable
solutions. One is the Kirchhoff method — that the acoustic field at
any point on the surface can be approximated by the field that would
be on a plane tangent to that point on the surface. Another is the
Fraunhofer phase approximation — in the expansion of the phase of an
exponential only the linear terms in the spatial coordinate system
are retained. Claude Horton has shown the necessity in certain geo-
metrical situations where also quadratic terms should be retained to
yield the familiar Fresnel approximation (Melton and Horton, 1970).
Inherent in Eckart's approximation for the normal derivative of the
reradiated pressure is the implication of a surface with zero slope.

It is also inherently assumed that there is no shadowing such that each
facet on the sea surface is completely insonified. Brekhovskikh

(1952) has given some restrictions on angle of incidence, surface
curvature, and acoustic wavelength for complete insonifications of

the surface irregularities. Research, both theoretical and experi-
mental, at the Applied Research Laboratory, University of Texas, into
the validity of the above approximations and assumptions has been

extremely useful to other investigators in surface scattering.

We can take the same approach used by Eckart and solve the re-

sulting equation for a traveling sinusoidal surface given by

G (xpyv,e) 5= kh cos las - kx cos a - ky sin a] (4)

With the proviso that the surface slopes are small and other geometric
approximations are met, the solution can be compared to experimental
results. Roderick (1968, 1969) conducted small-scale tank experiments
in which acoustic waves were scattered from a traveling sinusoidal
surface created by an electrical-mechanical wave generator. Wave

heights and surface wave lengths were accurately measured over the

33]!

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

insonified area, and conditions could be generated that modeled low-
frequency sound propagation interacting with the gravity waves of the
sea surface. Predictions were made for the normalized pressare re-
flected and scattered from a traveling sinusoidal surface of angular
frequency Woe wave height h, and wave number k. The wave is propagating
in a direction that makes an angle @ with a vertical plan containing

the angle of incidence and reflection.

An interesting result is observed for the scattered sound: the
spectrum of the reradiation contains upper and lower sidebands posi-
tioned symmetrically about the transmitted frequency w and displaced
from w by multiples of the surface frequency. The amplitudes of the
frequency components are given by Bessel functions of the first kind
and of order n. The argument of the Bessel functions are dependent
on the angles of incidence and scatter, wave height, and acoustic wave

number. These relationships are summarized in the following equation:

When the surface wave length is much larger than the acoustic
wave length, most of the acoustic energy is scattered close to the
specular direction, and it is not possible to resolve the specularly
scattered signal (see Figure 3). The acoustic energy is scattered
in space in selected directions determined by the familiar diffraction-
grating equation of order n. (This same equation appeared in Flatté's
talk (these Proceedings) during the discussion of the interaction of

internal waves and acoustic fields.) The carrier frequency is

33/2

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

WVdOVId DONIYXLLVYOS Ava

(si -@ uss) _uls = “p

(xy-45m)soo y29

°€ eaNnbty

333

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

reflected specularly and for small acoustic-—to-surface wavelength
ratios the energy is scattered in space symmetrically about the
specular direction. For the direction of surface wave propagation
shown, you would receive a down-Doppler in the backscatter direction.
On the other side of the specular direction you would receive an up-
Doppler. Reversing the direction of surface wave propagation, you
would also reverse the directions of the Doppler shifts, e.g., an up-

Doppler would be received in the backscatter direction.

: : 3 th : :
The scattering directions of the n order sidebands are given

by the angles 8, and 943

A
n a
8, i oil ~ cos 6 A aa ee)
1 s
vA
0, = e-n ia a (7p)

where 8. is the angle measured from the normal to the surface and is
an angle of elevation, and o, is an azimuthal angle measured from the
vertical plane containing the angle of incidence and reflection. The
directions in which the energy is scattered are functions of the ratio
of the acoustic-to-surface wavelength, the angle of incidence, and the

direction of surface wave propagation.

The effect on the azimuthal scattering angle 8, by the direction
of surface wave propagation and the acoustic-to-surface wavelength
ratio is shown in Figure 4. For small ratios the scattering is close
to the specular direction, and it is not possible to resolve only the
specular component at the transmitted frequency. The Doppler fre-

quencies are scattered on each side of the specular direction.

334

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

OILWY HLONATSAVYM AOVAIYNS-OL-OILSNOOV
SOASURA AIONVY ONINALLVOS TWHLAWIZY ‘py eanbtga

006 = re)

335

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

As mentioned previously, low-amplitude sinusoidal waves were
created on the water surface of an anechoic tank. A micrometer was
supported above the surface and lowered to measure the wave height.
The surface wavelength was measured from two wave-height sensors and
the results compared well to the dispersion equation for gravity
waves. With knowledge of acoustic parameters and geometry, predic-
tions can be made for the scattered field. The spectrum on the upper
left of Figure 5 is the amplitude frequency spectrum of an acoustic
signal reflected from a calm surface. The other three spectra are
the returns, measured in the specular direction, from a surface of
4.5 Hz with waves propagating in directions equivalent to up-wind,
down-wind, and cross-wind. The wave heights for the three cases
were the same. Note that in each spectrum the sideband frequencies
are symmetrical about the carrier and displaced from the carrier by
the surface frequency, 4.5 Hz. The sideband frequencies for the 4.5-
Hz surface waves were scattered within 3 degrees of the specular
direction and the energies in the sideband frequencies are identical

regardless of the direction of surface wave propagation.

Looking only in the specular direction, it is not possible to
observe the effects of the spatial scattering of the sidebands (see
Figure 6). Placing a hydrophone 10 degrees off specular and toward
the backscatter direction, the Doppler shift was measured at a sur-
face frequency of 6.0 Hz. The surface waves were propagating ina
direction equivalent to up-wind and, as expected, an up-Doppler was
obtained. There are no lower sideband frequencies and the spectrum
consists of the first- and second-order sidebands. Reversing the
direction of the surface waves, we obtain a down-Doppler, as shown
in the bottom spectrum. The received signal consists of just the
lower sidebands. The wave heights for these two cases were not the

same.

336

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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AIP/ ZH L'€:31IWDS ADN3INOSAS
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UND FROM THE SEA SURFACE

FORWARD SCATTERED LOW-FREQUENCY SO

RODERICK:

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(ap) JGNLNdWYV 3AILV1394

338

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

As mentioned previously, Parkins predicted the Doppler spectrum
of an acoustic signal reradiated from the ocean surface using physical-
optics techniques. He uses basically the same formulation of the Helm-
holtz Integral as Eckart, but also includes a slope correction term
originally formulated by Brekhovskikh and Isakovich (1952). The
Doppler spectrum is obtained through the Fourier transform of the
reradiated autocovariance function. For low frequencies, the Doppler
spectrum consists of a specularly reflected component at the fre-
quency of the incident radiation and two scattered components that
are Doppler shifted symmetrically about the incident frequency. The
magnitude of the deviation of the sideband frequencies is the same
and depends on the angles of incidence and observation relative to

the wind direction and also on the incident frequency.

In 1970, Ben Cron and I did some experimental measurements
(Roderick and Cron, 1970) of the Doppler spectrum. An acoustic path
that included a surface reflection at a grazing angle of 7 degrees
was used between the DOSS array and the TVA. The DOSS array consists
of two magnetostrictive scrolls which generate 750 and 1,500 Hertz
in the water. The TVA consists of 40 hydrophones positioned in a
vertical array and was used to beamform to receive the surface
reflected signals and minimize undesirable multipaths. The spectrum
in the upper left of Figure 7 represents the spectrum of the signal
incident on the surface. Before we go any further, the analysis was
done on a real-time spectrum analyzer in a frequency range of zero to
five Hertz. The acoustic signals were bandshifted to a center fre-
quency of 2.5 Hertz. On the upper right of Figure 7 is a spectrum
of the wave height measured at Argus Island using a resistive wave
staff. The wave height was measured at the same time as the acoustic
reflections from the surface and at a location which was 30 miles west

of the isonified area. The spectrum in the lower left is from the

33:9

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

WIND SPEED: 25 KNOTS

INCIDENT OCEAN
SPECTRUM SPECTRUM

=
=
an
ate
te
a
a

Ea |

Be
nt TE, TT
Te i

2.5 Hz 0.5 Hz

REFLECTED SPECTRA
750 HZ 1500 Hz

RELATIVE AMPLITUDE (db)

eee ea

Eat
Na EVI IM LN

co PR INTRE
I LM L A IVA WIE VT
ey \ ANT T We | | Wl
AE

25 Hz
FREQ. SCALE: 0.1 Hz/ DIV

Figure 7. RELATED SPECTRA

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

750 Hertz transmitted signal. Note that the sidebands are displaced
equally from the carrier. The spectrum obtained at the transmitted
frequency of 1,500 Hertz shows significantly more energy in the side-

band frequencies. (All vertical scales are 5 dB per division.)

The spectra shown in Figure 8 are for two consecutive pulses
reflected from the surface and separated in time by 3 minutes. Note
again that the sideband frequencies are displaced symmetrically about
the carrier and peaked at the frequency of maximum energy on the

surface.

For a wind speed of 35 knots, the reflected spectra (Figure 9)
have their first-order sidebands peaking at approximately 0.07 Hertz.
It can be seen that the carrier frequency is suppressed for the 1,500-
Hertz case; thus, almost all the received energy is contained in the
scattered frequency components. The ocean spectra recorded for this

wind speed of 35 knots are also peaked at 0.07 Hertz.

In a recent JASA article, Vertner Brown and George Frisk (1974)
reported on Doppler spectrum measurements conducted in the open
ocean in the frequency range of 100 to 500 Hertz. The statistics of
the sea surface were measured simultaneously with acoustic data by
a surface-sensing buoy. The acoustic spectra are compared with the
surface-wave spectra at each of the transmitted frequencies in Figure
10. For small surface roughness, the acoustic spectra contain the
discrete carrier frequency component with sidebands symmetrically
positioned about the carrier. For moderate roughness, marked
asymmetry in the acoustic spectra and strong spectral components

that are not prominent in the surface spectra are found.

Harry DeFerrari and Nghiem-Phu (1974) published the scattering

functions of various acoustic arrivals over a propagation path of

341

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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342

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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YdlddVo
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343

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE
— ACOUSTIC SPECTRUM
ie 100 Hz
| — ACOUSTIC SPECTRUM BI GHETE|
100 Hz -20+
—— SURFACE WAVE
1917-1927 SPECTRUM
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1944-1954 2147-2157
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I FREQUENCY (Hz) 0
Figure 10. DOPPLER SPECTRUM MEASUREMENTS

AT 100, 250, AND 500 Hz

(from Brown and Frisk, 1974)

344

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

7 nautical miles. The scattering function, as shown in Figure ll, is
a three-dimensional description of the transmission in terms of the
intensity, frequency, and delayed time of arrival. For surface-
reflected-bottom-reflected arrivals, the Doppler spectrum has side-
band frequencies at surface-wave frequencies. The scattering func-
tion was measured during typical summer conditions with onshore winds
of 10 to 15 knots. Asymmetrical sidebands were observed in some of

the scattering functions.

Asymmetrical sidebands can result when scattering occurs from
a multi-frequency surface (see Figure 12). Consider the case of a
two-frequency surface: the magnitude of each frequency component
is proportional to the product of two Bessel functions and the side-
band frequencies represent all possible combinations of the carrier
and multiples of each surface frequency. If the surface frequencies
are commensurable, then each sideband frequency is made up of a
vector summation of the individual terms. In general, this will
result in asymmetrical sidebands. The spectrum shown for surface
frequencies of 4 and 6 Hertz has asymmetrical sidebands at a dif-

ference frequency of 2 Hertz from the carrier.

The AFAR range has also been used (O'Brien, et al., 1974) to
measure the Doppler spectrum at different transmitted center fre-
quencies. At 600 Hertz (see Figure 13), the received energy is
coherent and predominately in the carrier frequency. As the fre-
quency increases, the Doppler spectrum consists of more and more
incoherent scattered acoustic energy. The sea state was 0.45 meters

rms.

In terms of the Doppler spectrum, we want to know what has been
done; where we are; and what needs to be done. I have tried to

illustrate the state of these affairs. We have made enough

345

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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346

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

FREQUENCY FREQUENCY
600 Hz 850 Hz

6 36
SA - 3 -.2°>) “OR 52 sed =A 33) =.2=1) 0...) 2s

Hz z
-12 FREQUENCY FREQUENCY
1300 Hz 1800 Hz
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Hz Hz

Figure 13. DOPPLER SPECTRA AT SELECTED FREQUENCIES
(from O'Brien, Pearson and Freese, 1974)

348

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

measurements of the relative energy in the Doppler spectrum with
supporting oceanographic data. You could fill a bookcase with the
various theoretical models that are available in predicting scattered
sound from the sea surface. However, I cannot determine whether the
present models are adequate enough to produce the absolute value of

the coherent as well as incoherent energy of the Doppler spectrum.

To my knowledge, there has not been any deep ocean measurement of

the absolute value of the Doppler spectrum. Lastly, there are
measured asymmetrical sidebands obtained by various experimentalists —

theoretical predictions should be made to compare to these results.

In the remaining time, I would like to give a brief overview of
an experimental program in measuring bistatic reverberation from the
sea surface presently being conducted in the Block Island-Fishers
Island Range. A parametric source (Figure 14) is being used to
generate a narrow beam of acoustic energy incident on the sea surface.
The source characteristics of wide bandwidth, narrow beam width, and
no sidelobes result from the array of virtual sources created by
the nonlinear interaction of the acoustic waves in the water medium.
The reverberation from the surface is received on the vertical array
of transducers and the information is cabled to shore. As you may be
aware, the reverberation Doppler spectrum is a function of the sea-
surface directional wave spectrum. To obtain an estimate of this
spectrum, an array of five upward-looking transducers is used to
measure the wave height as a function of time. The near-field
characteristics of the transducers are used to isonify a small spot
on the surface. The transducers are positioned to obtain equi-

spaced cross-power spectral-density functions.

A typical beampattern of the parametric source measured at a

difference frequency of 7 kHz is shown in Figure 15. Note the absence

349

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

WALSAS DNIYASVAW OILVLSId “pt eanhbta

O. INOHdOYGAH
ms

3DyNOS
<a DIVLIWVAVd

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350

FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

RODERICK:

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Ol2

Sjoyll

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

of sidelobes, and the narrow beamwidth of the major lobe. A 10-inch
diameter piston transducer operating at a reasonable frequency of

approximately 250 kHz was used as a source.

Figure 16 displays recordings of wave height versus time measured
at five discrete points on the sea surface. The sensors are posi-
tioned in a line array and, if you look closely, you can see the phase

relationship as the surface waves propagate across the array.

352

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

YN
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353

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

REFERENCES

Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic
Waves from Rough Surfaces, The Macmillan Co., New York, 1963.

Brekhovskikh, L. M., and M. A. Isakovich, "Diffraction of Waves from
a Rough Surface," translated from the Russian by R. N. Goss,
Naval Electronics Laboratory, Rpt. No. 14888, 1952.

Brown, M. V., and G. V. Frisk, "Frequency smearing of sound forward-
scattered from the ocean surface," J. Acoust. Soc. Am. 55:744-749,
1974.

DeFerrari, H. A., and Lan Nghiem-Phu, "Scattering function measure-
ments for a 7-nm propagation range in the Florida Straits,"
J. Acousté. Soc. Am. 56:47-—52, 1974.

Eckart, C., "The scattering of sound from the sea surface," J. Acoust.
Soc. Am. 25:556-570, 1953.

Ellinthorpe, A. W., "Frequency smearing on undersea acoustic paths
with fixed end points," J. of Underwater Acoust., 16, 427-435, 1966.

Mackenzie, K. V., "Long-range shallow-water bottom reverberation,"
J. Acoust. Soc. Am. 34:62-66, 1962.

Marsh, H. W., and E. Y. T. Kuo, "Further results on sound scattering
by the sea surface," AVCO Marine Electronics Office, 1965.

Melton, D. R., and C. W. Horton, Sr., “Importance of the Fresnel
correction scattering from a rough surface: I. Phase and Amplitude
Fluctuations," J. Acoust. Soc. Am. 47:290, 1970.

O'Brien, G. J., J. H. Pearson, and H. A. Freese, "Surface-induced
frequency smears on forward scatter paths," Naval Underwater
Systems Center, TE-68-74, 1974.

Parkins, B. E., "Scattering from the time-varying surface of the ocean,"
J. Acoust. Soc. Am. 42:1262-1267, 1967.

Roderick, W. I., "Acoustic spectra of specular and near specular
scattering from a three-dimensional traveling sinusoidal surface,"
paper presented to the 78th Meeting of the Acoustical Society of
America, 1968.

354

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

Roderick, W. I., "Frequency spectra of forward-scattered underwater
sound from a traveling sinusoidal surface," Navy Underwater Sound
lab., Rot. Now 988, L969)

Roderick, W. I., and B. F. Cron, "Frequency spectra of forward scattered
sound from the ocean surface," J. Acoust. Soc. Am. 48:759-766, 1970.

DISCUSSION

Dr. P. W. Smith (Bolt, Beranek, and Newman, Inc.): The example
or the explanation you gave of the asymmetrical sidebands suggested to
me that they were a peculiarity of the signal reflection that would

not result if you averaged over many reflections.

Mr. Roderick: The spectrum I showed was a single time record
with only 2 degrees of freedom. In other spectra that have been
ensemble-averaged (including Brown's, DeFerrari's, and some results as
seen on the BIFI range), one sideband can be down perhaps 5 or 6 GB.
You have a good statistical confidence in the spectrum due to the

large number of degrees of freedom.

Dr. Ira Dyer (Department of Ocean Engineering, Massachusetts
Institute of Technology): Bill, what hope would you hold out in using
an acoustic system for getting the wave spectra, wave number spectra as

well, of the ocean surface or any other rough scatterer?

Mr. Roderick: I think this can be done — with some qualifica-
tions. Bob Williams did his Ph.D. thesis on measuring the gravity

wave spectrum based on acoustic results.

Unfortunately, he had a horrendous problem. His acoustic path
involved many surface reflections and he had very poor control over
geometry. I think he only got fair results, mostly because of the

experimental setup. I think it can be done.

B55)

RODERICK: FORWARD SCATTERED LOW-FREQUENCY SOUND FROM THE SEA SURFACE

Dr. Walter W. Munk (Institute of Geophysics and Planetary Physics,
University of California at San Diego): The corresponding electro-
magnetic scattering problem which is very similar has recently been
attempted — a real comparison between the backscattered energy with
the backscattering geometry, and that computed from wave measurements.
Even an absolute comparison was attempted, and it came out 2 or 3 dB

off on the absolute comparison, but quite well on the relative.

356

COHERENCE

Theodore G. Birdsall

Cooley Electronics Laboratory
University of Michigan

My theme is that much of the randomness in underwater
acoustics is not "inherent" randomness, but rather is the
manifestation of complicated deterministic phenomena.

INTRODUCTION

What is "coherence" and, more important, what good is "coherence?"
Definitions can be "Sticky." The pun is intentional. Coherent means

to consist of parts that stick together, that are logically consistent.

In various disciplines coherence has taken on special meanings,
often related to techniques of quantifying (measuring) the degree of
coherence. Quantification is necessary, but it can carry hidden
assumptions that can confuse and even impede progress. For example,
a correlation coefficient and its decay in time or space is most
appropriate for first order Markov processes. The more sophisticated
"coherence function" is most useful for wide-sense stationary

Gaussian random processes.

This paper is concerned with underwater acoustic propagation,
with "coherence" meaning the consistency of reception across time,
frequency, and space. The viewpoint is that of a signal processor,
concerned with extracting information from acoustic receptions. This
means information about propagation, or extracting operational infor-

mation about targets or false targets.

Signal coherence is most important in weak signal situations;
that is, when the signal power is small compared to the noise power

or the signal's own reverberation power. The sub-discipline known as

Shy)

BIRDSALL: COHERENCE

"signal detection theory" is a study of precisely how uncertainty

about signal waveform and uncertainty about the noise characteristics,
together with interfering power, affect signal-processing design. Per-
haps more important, it studies how uncertainty and noise combine to
limit ultimate performance in signal detection and information extrac-
tion. The theoretical results are complicated, but there are some guid-

ing general principles.

One such general principle is that the effective signal-to-noise
ratio after processing will fall off sharply at low signal-to-noise
inputs. Just where this 'suppression effect' cuts in depends on the
degree of uncertainty about the signal and noise characteristics. When
the input is well above the 'knee' most reasonable processors work
about equally well, and signal uncertainty is not very important. The
other side of the coin is that processors can be designed to dig infor-
mation out of weak receptions only if there is substantial knowledge
of signal characteristics and this knowledge is used. Said again, if
detailed signal knowledge is available and is used, it may mean tens-

of-dBs of processing gain.

BRIEF HISTORICAL REVIEW

Twenty years ago many U.S. propagation people doubted that there
was sufficient stability in signals propagated over long distances to
support detailed signal knowledge. There were notable exceptions.
Project Artemis of Hudson Labs was a courageous step forward in investi-
gating propagation stability as seen at a very large receiving array.
One must qualify all experimental results as being specific to the areas
and frequencies studied. For Artemis that means for the Atlantic area
south of Bermuda and in the neighborhood of 400 Hz. Artemis established
the predominance of RSR (refracted, surface-reflected) paths for long-
range propagation, and the importance of the bottom topography and

local internal waves in the neighborhood of a slope mounted array.

358

BIRDSALL: COHERENCE

The measures of space coherence were primarily (1) pulse time of arrival,
and (2) the linear correlation coefficient estimated from clipped pro-
cessing. That correlation would drop sharply to one-half, and then fall
off slowly as the spacing increased. Tracking and prediction of non-
plane wave fronts was limited by the speed and size of the available

computers.

In the same time frame the NEL studies with pseudo-noise trans-
missions, controlled transmissions that covered over one-third of an

octave, showed substantial waveform repeatability.

Almost all experiments over long ranges involve at least one moving
platform. Great care has been taken to reduce the fluctuation of the
platform, through using submarines and drifting ships. This care is
influenced by the experimenter's opinion of the stability of the medium.
There is little to be gained by reducing the platform instability effects
far below the effects that will be caused by the medium. As instrumen-
tation improves we often repeat the old experiments and get different
results. In a drifting-ship to bottomed-receiver experiment in 1963,
using CW (a 420 Hz tone), across the Straits of Florida, a frequency
stability of 4 millihertz was observed. That stability was comparable
to the frequency source stability and to the ship station keeping.
Subsequent fixed-site experiments with improved sources confirmed this
stability in the Straits of Florida and over the old Artemis range in

the Atlantic.

This millihertz frequency stability is a nice example of the com-
plexity of 'coherence'. It does not mean that the received signal looks
like a pure tone. The signal shows substantial amplitude fluctuations
and some phase fluctuations, which are now recognized as the effect of
forward scattered surface reverberation. Mother Nature thoughtfully

arranged for this reverb to lie in frequency sidebands some 50 to 500

Sj5)")

BIRDSALL: COHERENCE

millihertz to either side of the carrier frequency. That reverberation
is part of the 'incoherent' part of the reception; its lack of struc-
ture makes it much less useful than the stable signal line. Measure

its power and then filter it out; once removed the remainder is the
signal that possesses the millihertz stability. That is lesson number
one: partially coherent signals may sometimes be separated into coherent
and incoherent parts. The separation increases our understanding of
propagation. The coherent part is operationally much more effective

at low signal-to-noise for detection and identification, and worthy of

further study.

Studies of the isolated stable line showed that life is really not
simple. In a multipath situation - and that is the usual situation
for many of us - it is common for the amplitude of the line to vary
substantially, while the phase of the line (or its instantaneous fre-
quency) has such slow variations that it reflects tidal and internal-
wave behavior. If one models 'paths' as slowly and independently vary-
ing, the model disagrees. However, if one models 'paths' as slowly
and dependently varying, reacting to the same global temperature
variations, then the model begins to fit. That brings in lesson num-
ber two: the propagation may be coherent, that is, complicated but
logically consistent and dependent of the same variations, and yet
yield some measurements that appear to be incoherent. It is up to the
scientist and the sonar designer to seek, recognize, and capitalize on

whatever 'coherence' nature provides.

NON-MARKOV COHERENCE

There is a natural tendency to believe that 'coherence' should
behave in a Markov fashion in all dimensions. We seek coherence dis-
tances, coherence time constants, coherence bandwidths. We ask ‘how
far apart do receptions have to be before coherence drops tO one-over-e?'

as if that just has to be an intelligent question.

360

BIRDSALL: COHERENCE

Does the consistency of propagation break down as signal frequen-
cies are separated? It is common to experience different fades at sig-
nals just a few Hertz apart. I would like to cite just one study to
indicate that the behavior in frequency is more coherent, and more
complicated, than a Markov process. Single-path loss measurements were
made at 61 frequencies spaced 5/6 Hz apart, covering the regime from
395 Hz to 445 Hz over a period of 7 hours. The transmission was over
43 miles across the Straits of Florida. The loss contours as a func-
tion of frequency and time show a lot of pattern; I hope enough to
encourage studies that go after the whole surface, and enough to dis-
courage attempts at determining a correlation bandwidth. The frequency
deviation plot for the same data shows major peaks of the order of
one millihertz wide. Low-magnitude broadband ripples and changes
slide across frequencies in time; however, the entire band has a rea-
sonable unity. Of course it is only about one-sixth of an octave, but
that is all many sonars (active) cover. Incidentally, these data were

taken with nine-foot seas overhead.

Correlation time-constants for multipath propagation are another
popular concept. In some locations it may be a valid description of
multipath behavior. Again, I would like to cite one study to indi-
cate that multipath propagation may be more coherent than suspected,
but that much careful work will have to be done to discover the
coherent parts and to use them. The data spans one day, and used a
continuous transmission designed to yield the same time resolution as
a 20 millisecond pulse repeated every 1.2 seconds (but with 18 dB more
processing potential). The data taken in November 1971 show a dominant
30-millisecond arrival alternately merging and contrasting with a
following weaker arrival. This routine structure shows a dramatic

change both in the duration of the arrival and in the phase pattern.

361

BIRDSALL: COHERENCE

The arrival frequency shifted 0.5 millihertz and 12 hours later changed
again by 0.8 millihertz. Even the weak trailing arrivals show re-
peatable phase often lasting for hours. There is strong evidence of
consistence and pattern, but it is complicated, and one time-constant

does not describe this type of data.

HALF-TIME SUMMARY

Coherence is a complicated subject, but worth pursuing because of
the potential gain in apt signal processing at low signal-to-noise
ratios. Coherent propagation may lead to complex receptions which must
be sorted out, and some physical measurements will appear to be much
more coherent than others. The lack of regularity in one class of
measurement does not imply incoherent propagation, and simple measure-

ments of correlation may be deceptively uninformative.

In the second half of this paper a model of propagation from a
submerged moving source will be presented to show how a complicated
and apparently incoherent signal may be received even though the pro-
pagation itself is totally coherent, totally deterministic. The model
agrees well with measurements in many respects, but I beg your in-
dulgence for leaving that to another paper. The purpose of this pre-
sentation is to emphasize my theme: Much of the randomness in under-
water acoustics is not inherent randomness, but is rather a manifesta-

tion of complicated deterministic phenomena.

AN EXAMPLE MODEL

Picture a deep ocean with a single classical sound speed profile
that applies everywhere, and a fixed source at 150 meters depth and
600 km from a deep receiver. The numbers are purely for example

sake. This source emits a steady pure tone, let us say at 250 Hz.

362

BIRDSALL: COHERENCE

All this is said to formulate a multipath situation. Paths will appear
in pairs, with one path of each pair leaving the source at some de-
pression angle, and the other path of the pair leaving the source at

an elevation angle of almost equal magnitude. The difference in the
absolute value of the angles is quite small, of the order of .1 degree.
At the assumed long range there will be several such pairs, and the
reception will be the vector sum over all paths with their travel times

and losses; it will be a 250 Hz signal.

Now let the source open range at 6 knots. Grossly speaking this
will cause a 0.5-Hz Doppler shift. Speaking more carefully, there will
be a 0.5*cos(angle) shift. Consider one pair of paths. If their
angles differ by 0.1 degree, their Doppler shifts will differ by 0.152
millihertz. That's not much, but therein lies the key number in this
model. Since the difference is so small no current receiver will
separate them, and their sum will appear as a single frequency, with
an apparent fade rate of 109 minutes. (Of course I am thinking about
much shorter observation times than 109 minutes.) The conclusion is
that the 'insignificant' differential Doppler will have almost no effect
on the measured frequency, but has a substantial effect on the measured

amplitude.

There will be a number of ray-path pairs. For simplicity assume
that a hypothetical analyzer can isolate two pairs in one narrow filter
with relative frequencies and amplitudes as listed below:

rel. freq. -.0100 =.0092 O -0005
rel. amp. (dB) Pal P-3 P P-2

The amplitude, linear with pressure, will show rapid fluctuations
with a period of 100 seconds separating deep fades about every 13
minutes. A phase tracker shows the instantaneous frequency of the

reception fluctuating wildly at the 100-second rate, but using the

363

BIRDSALL: COHERENCE

phase tracker's average slope yields frequency estimates which have a
"capture' behavior, locking onto the frequency of the momentarily

largest line pair.

This example was based on only 2 path pairs, and the 2.7 km range
change was ignored in the calculations to emphasize the simple beat.
Add a few more path pairs, take into account the opening range and
the attendant path and angle changes, and you will obtain a complica-
ted fade pattern, and a frequency estimate that hops around. Is that
bad? No, that's good! It's a basic part of propagation, and it occurs
in every test and every operational situation with a moving source at
long range and not on the surface. The fluctuations were coherent.

There was no randomness in the example.

That concludes coherence lesson number three: not everything
that varies is a random variable, nor do fluctuating receptions imply

incoherent propagation.

APOLOGY

My brief history omitted reference to almost everyone, and I owe
an apology both to the researchers and to this audience if they were

expecting a scholarly review.

364

FLUCTUATIONS: AN OVERVIEW

Ira Dyer

Department of Ocean Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts

The study of fluctuations is hampered by a diversity of
definitions and language. I propose that theories and
experiments be distinguished as to 1) Averaging Algo-
rithms, 2) Source/Receiver Motion, and 3) Ocean Dynamics.
Confusions in the interpretation or comparison of recent

results can be reduced thereby. Several results are
discussed within this framework.

More particularly, a format for studying short-time-

averaged amplitude (envelope) fluctuations is proposed.
Envelope statistics for phase random processes are well
known for sample sets with fixed mean; these statistics
change significantly for sample sets with varying mean,

as is often the case for sets extending over time dura-
tions and spatial extents involved in detection by typical
sonars. The format includes metrics such as the fluctu-

ation (fading) period, the fluctuation (interference)

scale, and more generally their corresponding spectra.

Some recent results are discussed in these terms.

There is no clear, accepted way to distinguish between various

categories of fluctuations. This paper addresses, therefore, the

question of time series that may be observed in the ocean;

equally to spatial series that may also be observed in the
the study of coherence. Figure 1 defines certain terms of

to time series; a comparable set of definitions exists for

spatial series.

it applies
ocean in
importance

the

Two quantities are typically of interest: the amplitude, which

may be the decibel level, A (or the intensity, I, or the RMS value,

|p|), and the phase, 0, which could just as well be the tilt angle

or the bearing angle.

365

AN OVERVIEW

FLUCTUATIONS :

DYER:

Sdidds AWIL NI SaadLdwvaivd

*—T oazanbtg

<
7V>

———

< (1+3) V (9) V>

366

DYER: FLUCTUATIONS: AN OVERVIEW

The time series, A(t), has a probability density, P(A), anda
spread, AA(Q) called the fading range. This can be defined in terms
of the cumulative probability distribution, and in the subsequent
results it will correspond to the fading range where 2.5 percent of
the lowest levels and 2.5 percent of the highest levels are dis-

carded.

Similarly, a period, Ths can be defined as the time between
crossings of a level Ns going in the same direction. The period is,

in fact, a function of that level.

The crossing period T is therefore related to the crossing

rate T1

A
of the slope of the curve times the joint probability density of

which is given by the integral of the absolute magnitude

that particular level which it crosses and the slope itself. In the
subsequent results, the periods will correspond to the crossing level
that is the mean of the time series. In case of phase, for example,

that mean will often be zero.

A third measure, the autocorrelation, p, is defined in the usual
way, and the time, t, will correspond to the 1/e point on that auto-

correlation function.

Figure 2 summarizes some of the knowledge acquired by those
working on the Eleuthera-Bermuda experiment. There are several
fading types. The very fast fading type extends from an averaging
time of perhaps ten times the period of the sound wave itself up to
approximately 15 seconds. A fading in this domain of averaging time
and record time — that is, the length of time we look at a record —
is usually incorporated within scattering theories and scattering

experiments.

367

: AN OVERVIEW

FLUCTUATIONS

DYER:

NVDVWNVT4 ‘WUWIO ‘OUHENISM ‘GYOANWLS ‘IuvuNgdaed
(0 = A ‘uu 0O0L = U ‘ZH OOF = F ‘epnuzeg/erzeyzNeET)
SOILSIYALOWYVHO IVYOdWEL °*Z eanbty

TOs 4390s aor

| (suotjenqonT 4 SHbutzeaaeds) | 292 85 3/0T
| | || | | qasez AzoA

uTUW GZ uTU Q8 KONG UTU G UTW SZ dp 0z AU «99S GT
4Seaq
K
ay ZT KO OT IU ZT ap ¢ eee uae
a7 eTpowurejUrL
ye is
Xp 0€ Ko OL Kp 0€ ap st EY, es
MOTS

LAT Ap OP

MoTS AZOA

|
ae

L L
edAy butpeg

368

DYER: FLUCTUATIONS: AN OVERVIEW

At the other extreme are very slow fades which have been
described as seasonal fluctuations. Perhaps the record time for
these is as long as 1 year and averaging times of 1 month or more

might be used.

Between these obvious extremes are other kinds of fading, per-
haps three or more, worth distinguishing between because they appear
to be associated with separate mechanisms. For example, fast fading
fluctuations tend to have a fading range of 20 dB in amplitude, a
period of about 25 minutes, and a decorrelation time of about 5
minutes. The phase, on the other hand, typically ranges over 5 cycles
with a 60- to 80-minute period, and perhaps a 25-minute decorrelation

time.

The intermediate fades yield rather different numbers. It should
be emphasized that this particular set of experiments is for one
frequency with fixed source and receiver locations. Hence, the only
motion that does occur is, in fact, the motion of the ocean. For the
intermediate fades, there is a definite period of 12.4 hours, with a
somewhat reduced fading range on amplitude and an increased fading

range on phase.

Figure 3 illustrates a possible mechanism for the fast-fading
case which leads to the general result that the fading period on
amplitude is related to the fading period on phase by the simple

ratio 1/90 *
rms

The argument proceeds from a modal interference picture which
treats the mean square pressure as a sum of sinusoids. Whenever the
phase difference between pairs of modes approaches twice the average
phase, an amplitude beat is generated. Hence, that amplitude beat

is given whenever the average phase is of the order of 1. Typical

369

AN OVERVIEW

FLUCTUATIONS :

DYER:

ONIGV4I-LSVd YOd WSINVHOAW ATAISsOd ‘“€ 9anbtgy

seTohko se Toko Suwa
iS OV 6, ~ Vi oste eve, S 8 oF ~ Vo
—- _¢/ m
Ae Ae fre
ne 9 ?3eeq aepnyztTTdue 92 ®& "'y qysebret
u Ww uw Ae rr
jon V g§ UOTRZOUNF MOTS “'W
4 G G sated Ct
Bd ,s00 + Wu 7809 ~ (] 7500, Shy cee d=
To
ae = 5 ==> b pue d
ap ee)
Ae Z
A ecm qZeyy yons u pue w sepouw ppe pue oueuU
) ap.)
ols : AE [
(syuqed Abzsusa Tenbs) z('9uts 7) + z('esoD Zz) » <zd>

HONEYATAALNI TVAOW

370

DYER: FLUCTUATIONS: AN OVERVIEW

amplitude periods are then on the order of 1.5 divided by the number
of fade cycles. Referring back to Figure 2, for fast fading, T,/A6 ad
16 minutes, which is approximately two-thirds of the observed T,

of 25 minutes. The agreement for the decorrelation times is not as
good but still in the right ballpark. This argument does not seem

to work for the other fading types which suggests that for this partic-
ular frequency range and these particular choices of averaging times
and record lengths the underlying mechanisms may be different. The
intermediate fading rates may, in fact, be related closely to the
modal interference that is caused by internal-wave motion, and the
slow rates may be caused by planetary waves that have a different

kind of behavior with respect to the fading process.

The results of Figure 2 should not imply that a simple reduction
of experiments to a single number table is, in fact, possible.
Figure 4 displays results obtained by Stanford (1974), where two
amplitude time series are spatially separated by only 40 meters
vertically and 80 meters horizontally. The periods of fade, Dae
differ by a factor of 2, although the amplitude fade range is about

the same.

Figure 5 illustrates results obtained by Spindel et al. (1974),
and the experiment differs in two respects from that reported pre-
viously: the range is somewhat different; and, perhaps more sig-
nificantly, there is a drift velocity of about a third of a knot,
rather than a zero range rate. (Nonetheless, as will be shown sub-
sequently, this drift rate may he not too significant.) More impor-
tantly, these results show a tremendous depth dependence to the
fading. The fading range on phase is of the order of 26 cycles with
a period of 140 minutes for the deep receiver. For the shallow
receiver above the main sound channel, the fading range is 10 cycles

with a fading period of 64 minutes.

Sy7/Ak

AN OVERVIEW

FLUCTUATIONS :

DYER:

SSN IVA IVYOdWHL JO AONEGNAdad IVILVdS ‘pp 9anbty

RG zojyue :902n0
(PLOT) “PACSUe7S : co Th $M ost = Sel oz Ol 06 SZ oF

(Stel se eels a Sape fee sli Tv el Tr

uTU ston —

uTu op’, —>

‘dos [ej,uoztTzoy wogg

‘des [ePOTIASA WOOP

(A) 30NLNéWY

PHASE (cycles)

(cycles)

PHASE

DYER: FLUCTUATIONS; AN OVERVIEW

=—1500m depth

se
120

“
100

ere INQWWAL) Tohige! Tovl40 min

<3 305m depth
AOV1LO cyc T,v64 min

ts 4
120 140

ae
100

Source: Spindel, Porter and
Jaffe

Figure 5. SPATIAL DEPENDENCE OF TEMPORAL VALUES

373

DYER: FLUCTUATIONS: AN OVERVIEW

These results indicate that it is impossible to state ona
single-number basis what the fading parameters are. Nonetheless, it
is possible to discuss trends in these data and to try to understand

why changes such as these occur.

Figure 6 shows results taken under more or less comparable con-
ditions, where the decorrelation time for amplitude, TA, is plotted
versus the carrier frequency. There were three experiments: one by
Nichols and Young (1968) at about 270 Hertz, DeFerrari's (1974)
(which was included in Figure 2) at about 400 Hertz, and Webb and
Tucker (1970) at about 800 Hertz. The decorrelation time seems to be
reasonably described by something that is intuitively appealing —
namely, that the frequency times time is a constant approximated by
1800 (when the frequency is in Hertz and the decorrelation time is in
minutes). This suggests, for example, that at 100 Hertz the decorre-

lation time may be as long as 18 minutes.

The previous results have been attributed to effects of ocean
dynamics. Figure 7 addresses the question: What about making
measurements with moving platforms? For moving platforms, the notion
of a spatial scan is introduced. There are spatial fluctuations in
the acoustic field if the ocean is considered completely stationary
or "frozen." These fluctuations are described in terms of a correla-
tion coefficient relating changes in range and changes in depth. A
rangewise scale, hos and a depthwise scale, Lo are then defined in

terms of the 1/e points in the correlation coefficient.

Preston Smith proposed a theory which may have a direct bear-
ing on these measures. In the second case of an isogradient duct,
the radial or the rangewise scale was found to be proportional to
the wavelength divided by the square of an angle, @,. This angle

Q
is, in fact, the angle that encloses all the refracted rays that are

374

min

TEIN

a)
ZN

Figure 6.

DYER: FLUCTUATIONS: AN OVERVIEW

1000

Nichols and Young

DeFerrari

Webb and Tucker

DECORRELATION TIME AND CARRIER FREQUENCY
FAST FADING (Ocean Dynamics)

AN OVERVIEW

FLUCTUATIONS:

DYER:

(suoTzenqzonTy ueeoQ uezolz,4)
AYNLONULS GIAIA GNNOS JO SNOILVIYVA IVILVdS "2 emnbta

Vy < Ve Pao = Vi :potied oste
oT
A
Soa, Toh
%
(ueess0 uezorzz e HuTuUedS UT) AWIL NOILWISYYOORAa
29 be . a
aad y°O 1% % Load LNATAGWYSOSsI
eee
(PTETF
punos Le
pextu Uae Tet. Sab mii ae
-TT2eM)
>U3TWS
a = 25 eae = Lond adaadSOSI
“l= = (oor =
Z 4
2a (EG S.2.0,).0

e/T = (0’ %)9 (zy ‘ay) 9

376

DYER: FLUCTUATIONS: AN OVERVIEW

trapped in the duct. The constant of proportionality involves param-

eters such as the depth of the receiver, z and the depth of the

ae
uct. «Di

Using this scale, or whatever scale is appropriate to the problem
of interest, the decorrelation time is determined by the scale length

and the range rate. When making measurements with a moving platform,

the impact of this time on the measurements must be addressed.

Notice that the decorrelation time for platform motion and ocean
dynamics is proportional to the wavelength, as shown in Figure 6.
Hence, there should be a particular value of range rate which makes
the two decorrelation times equal. This speed depends on the path
geometry but for this case appears to be on the order of 3 to 5 knots.
That is, if the ocean is scanned at speeds substantially in excess of
3 to 5 knots, the fluctuation time scale will be governed by the
structure that exists in the ocean as if the ocean were standing
still and didn't have, say, internal waves. On the other hand, if
the ocean were scanned at speeds significantly less than a few knots
(for example, the one-third of a knot in drift used by Spindel et al.
(1974), the time scales may well be those associated with internal

waves or other ocean dynamics.

Some evidence for this is indicated by the NRL experiments where
range rates were 7 knots and horizontal scale lengths of 65 kilometers
were measured corresponding to the convergence zone spacings. A
closer examination of their spectral decomposition in wave number
(really interference scales) shows at 14 Hertz about a 9 kilometers
interference length which is roughly consistent with the results in

Figure 7.

The transmission-loss data shown in Figure 8 were supplied by

Earl Hays and are a good example of the effects of platform motion.

377,

DYER:

‘ $88 SESH | RRESEESS: ests ces VSSeeeSs
ae

FLUCTUATIONS:

ooeeeea see one SHES RES

AOUANGA

378

aS See
=

AN OVERVIEW

SSeeee- SS

fee =
SRSasss
eons Ses

Jeteeeeae

ZSSSESes
oe - aay

SAT LVIaa

TRANSMISSION LOSS DATA SHOWING EFFECTS OF PLATFORM MOTION

Figure 8.

DYER: FLUCTUATIONS: AN OVERVIEW

The experimental geometry consisted of a fixed string of receivers at
the depths indicated and a source closing from a range of 7 miles to
a closest-point-of-approach of 2 miles and continuing on out to about
7 miles. The source frequency is 130 Hertz and the source speed is

6 knots.

Notice that for the shallow (300-meter) receivers, periodicity
is about 10 minutes, whereas for the deeper receivers, between 2,000
and 3,000 meters, the periodicity is of the order of one or two
minutes. These time scales are consistent with the spatial scales
that exist in the ocean, as sampled by the various source-receiver
geometries. The temporal scales associated with internal-wave motion
in this geometry would lead to periods of 40 to 50 minutes for this
frequency. This is a good example of an experiment which yields
time scales that result from the structure of the acoustic field in

the ocean and not from the ocean dynamics.

The conclusions these various results suggest are that the ocean
can move and hence give some structure to received-signal fluctuations
and, also, the platforms can move resulting in additional fluctuation
structure. Both of these possibilities must be considered. In fact,
in many practical circumstances there are sources moving near 6 knots
and a technique is needed to combine situations where fluctuations
due to platform motion and ocean dynamics are comparable. No theory
adequately takes both into account. In fact, no theories adequately

treat either of the two separate mechanisms.

Figure 9 addresses a few more facets of the fluctuation problem.
In a data record which addresses fast fading but is also long enough
to include, for example, intermediate fading, variations appear in

the mean, of the individual fast-fade processes.

Uae

3719

AN OVERVIEW

FLUCTUATIONS :

DYER:

SOILSILWLS HAGNLITAWY °6 eAnbTYy

apr ire ~ (SzZ0°0) VV

ap 9°S¢ = 9/zuse = Vo

ap ctz == We6On OT. = he 1 Sorroneeon a

fe°7 = © bor 0h =] ce

ybtetTAey :|d|| a e Xie
Tetqusuodxe :f | a Sie =a dxe me (v)d

2: (QOuUsTeFASAULI Tepow) butpewq yseg

:butpez ysed

ae |

Surpey. GAvr tT polltetU] = een

380

DYER: FLUCTUATIONS: AN OVERVIEW

In many cases the probability density for amplitude, P(A), for
fast fading results from a phase random process and for the intensity
is exponential, while for the rms pressure it is Rayleigh. These are
equivalent statements and for the logarithmic distributions, the mean
is depressed by 2.5 decibels and the standard deviation is 5.6 deci-
bels. This result assumes enough paths (10 or more) to justify an

asymptotic limit.

For this distribution, the fading range (throwing away 5 percent
of the extremes) is about 21 dB, which is consistent with short

observation periods (under 2 hours) for the frequency of 400 Hertz.

The next step (Figure 10) is to describe the amplitude statistics
for a longer period of time than that which just corresponds to each

of the fast-fading segments.

If the probability densities of the individual processes are
known, the final probability density is found by averaging P(A) over
the variation of the mean itself, P(u,)- For example, for fast fading
alone, the probability density of the mean is a delta function, yield-
ing back the phase random process. For predominantly slow fading,
variations in the mean may be reasonably given by a Gaussian process
which generates a sufficiently large spread in the mean that the
probability density of the logarithmic amplitudes approach a Gaussian
distribution. There is evidence that, in fact, this occurs when data

are included from experiments over time periods of 30 to 40 days.

In the intermediate fading-rate case, the results are not so
easily described. Figure 11 shows results obtained by John Clark (1974)
and his colleagues last year, where the signal histograms (essentially

the probability densities) are plotted as a function of time. Each

381

: AN OVERVIEW

FLUCTUATIONS

DYER:

NVAW DNIAYVA YOU SOILSTILVLS

ebret szob as
Vig se uetssney (Vv) d

“OT aanbtga

*DNIGV4 MOIS

>DNIGVA LSVa

382

AN OVERVIEW

FLUCTUATIONS:

DYER:

SOILSILVLS ALVY ONIGVA ALVIGHWYALNI “TT eanbty

4AN3N93S AVA OML Y3d SINIOd 0882 ALN3W93S Avd OML Y3d SLNIOd 0882
S3I1dWVWS 3LNNIN 3NO Vivd SNISSIN S31 dWVS JLNNIW JNO

82/2) 92/01 JO SAVO Eb gig £/8

nm an

(ES eve eUess eae
: oo ES ;
- 9 va ~ PS ye 9
S NA ¢
> »

(8P) NOILVIAZO OYVONVLS

(GP)NOILVIAZO OYVONVIS

~
02! 5 aXe:

PD IR * |

(@P) SSOT NOISSINSNVYL NV3W (QP) SSOT NOISSINSNVYL NV 3N

00!

ol
2'ON VONWY3E |°ON VWONWY38

383

DYER: FLUCTUATIONS: AN OVERVIEW

probability density represents data taken over 2 days. The individual
densities are somewhat skewed as expected, but the individual densities
change with time. While the probability density of either entire

group has not been generated, the supposition is that it may approach
in and of itself a Gaussian distribution. Note that the means of the

distributions change with time as do the standard deviations.

The final figure (Figure 12) shows one possible way to treat
this. The solid curve corresponds to a single population consisting
of phase-random multipaths. It is skewed with the 2.5-dB depression
in the mean. (That is, the most probable value is 2.5 dB higher than
the mean value.) If seven such processes are added, uniformly spaced
with a spread in means of 6 dB, the resulting distribution is easily
integrated (since it consists of a sum of delta functions) and leads
to the dashed curve in Figure 12. Two things have happened: First,
the standard deviation has increased beyond 5.6 dB (the dotted curve
is broader than the solid curve); and, second, there is less skew and
peakedness in the distribution. In general, as the spread becomes

larger, the dashed curve becomes more and more GausSian in nature.

SUMMARY

In conclusion, there are many measures of fading. It is going
to be important to recognize various regimes of time for time series
and space for space series. It is equally important to indicate which
fluctuations are averaged out and which are included through the

length of the record.

The understanding of the sub-processes is quite far along; how-
ever, it is difficult if not impossible to include everything that
is observed. A more likely approach is to formulate very clear
statements about the particular process being investigated at a

particular time, recognizing the diverse underlying mechanisms.

384

AN OVERVIEW

FLUCTUATIONS :

DYER:

ONIGVA ALVIGEWAALNI YOd ATAWWXA ‘ZT eanbtd
1 G 0) é= Vase oS Ss

TeAeT ueow 32 gp

(ap z = 9 ‘ap 9 = Vnty)
suotjze[ndod ajezDstp paoceds ATWAOFTUN USASS op om ome

yzedt3zqtnu wopuerz-eseyd uotjzeqtndod ST HUTS cnn

385

DYER: FLUCTUATIONS: AN OVERVIEW

REFERENCES

Clark, J. G., and M. Kronengold, "Long-period fluctuations of CW
signals in deep and shallow water," J. Acoust. Soc. Am. 56:
LO7I=1083;, 1974.

DeFerrari, H. A., "Effects of horizontally varying internal wave-
fields on multipath interference for propagation through the
deep sound channel," J. Acoust. Soc. Am. 56:40-46, 1974.

Nichols; R. H., and H, oJ. Young, “Fluctuations in low-frequency
acoustic propagation in the ocean," J. Acoust. Soc. Am., 43:
716-722, 1968.

Spindel, R. C., R. J. Porter, and R. J. Jaffe, "Long range sound
fluctuations with drifting hydrophones," J. Acoust. Soc. Am. 56:
440-446, 1974.

Stanford, G. E., "Low frequency fluctuations of a CW signal in the
ocean," J. Acoust. Soc. Am. 55:968-977, 1974.

Webb, D. C., and M. J. Tucker, "Transmission Characteristics of the
SOFAR Channel, J. Acoust. Soc. Am. 48:767-769, 1970.

Weinberg, N. L., J. G. Clark, and R. P. Flanagan, "Internal tidal

influence on deep-ocean acoustic ray propagation," J. Acoust.
Soc. Am. 56:447-458, 1974.

386

SOUND PROPAGATION IN A RANDOM MEDIUM

Robert H. Mellen

New London Laboratory
Naval Underwater Systems Center
New London, Connecticut

For more than a decade we have been trying to identify
and measure the various factors within the water column
that contribute to the low-frequency attenuation in sound
channels. Experiments have been carried out in a number of
bodies of water, including fresh-water lakes, to study ef-
fects of temperature, salinity, and other environmental
factors. The results show an anomalous attenuation in sea-
water below 1 kHz in excess of the magnesium-sulfate relax-
ation contribution. A new relaxation mechanism involving
boron has been identified by Fisher and Yaeger. A second
anomaly is frequency-independent over considerable ranges
and is thought to arise from scattering by random vari-
ations in refractive index. Comparison of the scatter loss
estimated from random variations in sound-speed profiles
shows order-of-magnitude agreement with a wide range of
experimental results. Effects of the random component
of sound speed on spatial and temporal coherence within
the channel are discussed.

INTRODUCTION

For more than a decade we have been trying to identify the
sources and behavior of the various components within the water column
that contribute to the attenuation of low-frequency sound in the sea.
In 1967 we began a series of experiments designed to study the dif-
ferences in various bodies of water, both fresh and saline, of
different temperatures and other environmental factors. The map
in Figure 1 shows the regions that were studied and I would like to

discuss the results of these experiments (Browning and Thorp, 1972).

387

SOUND PROPAGATION IN A RANDOM MEDIUM

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Our technique was designed for simplicity of data analysis and
is represented pictorially in Figure 2. To eliminate boundary effects,
we made the experiments during the period when the surface tempera-
tures were sufficiently high to form a sound channel. SUS charges were
detonated on the channel axis as the transmitting ship opened range.
The signals were received by a hydrophone located on the channel axis

and recorded for later analysis.

Analysis was accomplished by using 1/3-octave filters and
measuring the total received energy arriving through refractive
paths. This is no problem since arrivals reflected at the boundaries
can usually be separated in time. If not, they can be ignored since
they are more severely attenuated at least at the frequencies of
interest. The results are plotted in decibels (corrected for cylin-
drical spreading vs range) for each of the filter frequencies. An
example is shown in Figure 3. Then by linear regression analysis,
we obtain the attenuation coefficient. The validity of the cylindrical-
spreading approximation and the neglect of bottom loss above a critical
frequency were checked by DiNapoli (1971) in his Fast Field Program

and will be discussed later by Browning (in these Proceedings).

ATTENUATION EXPERIMENTS

The saltwater results shown in Figure 4 together with earlier
work supported the conclusion of Thorp (1965) that the coefficients
below 1 kHz were anomalously high. The dashed line is the Marsh-
Schulkin curve that includes the MgSO, relaxation absorption. The
excess absorption below 1 kHz is greater than predicted by roughly
a factor of 10. Thorp fitted the anomaly to a relaxation formula

with a relaxation frequency of 1 kHz.

389

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The resulting attenuation shown in Figure 5 is a three-component

model consisting of:

e The fresh water viscous component
e The MgSO, relaxation component

e The anomaly which has recently been identified as a
second relaxation involving the boron content in sea-
water (Yaeger et al., 1973).

Thus, we may now say that all three components are absorptive and do

not involve scattering or other anomalies.

Several of the experiments do not follow the three-component
absorption model very well at all and Hudson Bay (Browning, 1971) is
one of those cases (see Figure 6). Since we have no reason to suspect
either the experiment or the absorption model, it is plausible that
the excess arises from some other mechanism. If we subtract the
theoretical from the experimental, we find that the excess attenu-
ation coefficient is a constant '0.04 dB/kyd over the frequency range.
This might suggest another relaxation below 100 Hz; however, this
hypothesis must be rejected for other reasons. A more likely cause

is forward scatter from inhomogeneities within the water columns.

As a first attempt to test this forward scattering hypothesis,
we have investigated the turbulent cell model of Chernov (1962). In
Figure 7 we see a plane wave progressing through a perturbed medium
where the refractive-index inhomogeneities are random, roughly
spherical, and have a scale size ao: The wavefront becomes corrugated

and the ray angles become randomly distributed. Energy is conserved.

In a sound channel, energy is normally trapped for all angles
less than some critical angle, Ona and leaks out for larger angles

(see Figure 8). Because of the diffusion by the inhomogeneities,

393

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

VISCOUS ABSORPTION

Mg SO,
RELAXATION

ANOMALY

ATTENUATION COEFFICIENT a@ (DB/KYD)

FREQUENCY (kHz)

Figure 5. ATTENUATION OF SOUND IN SEAWATER

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Figure 8.

THORP- FORMULA

DIFFUSION i

0.2 0.5 1.0 2.0 5.0 10
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SOUND-CHANNEL ATTENUATION (THEORETICAL)

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

the angular distribution of normally trapped rays increases so that
energy is continuously lost from the channel. The sound channel
attenuation model includes absorption, diffusion loss (Mellen et al.,
1974), and diffraction loss. Here the diffraction component is for

the first normal mode only and assumes an infinitely lossy bottom.

These data can be fitted empirically, as -shown in Figure 9, by
adding an extra loss independent of frequency. For the Hudson Bay
experiment, the excess is 0.04 dB/kyd while for the Gulf of Aden
(Browning et al., 1973) value, it is 0.02 dB/kyd. The MgSO , component
was corrected for temperature, -1.5°C for Hudson Bay and +15°C for
the Gulf of Aden. The most significant difference between the two
experiments is that the Thorp coefficient for the Gulf of Aden is
only 0.6 that for Hudson Bay, which suggests differences in boron

chemistry of the two bodies of water.

Once the possibility of a constant diffusion loss independent
of frequency was accepted, the results of Lake Superior (Browning
et al., 1968), shown in Figure 10, became clear. At first we had
guessed that the Thorp relaxation was common to both salt- and fresh

water, with only the MgSO, component missing in Lake Superior. It

4
was later found that the necessary boron content did not exist in

Superior which gave strong support to the scattering hypothesis.

We have used the term "independent of frequency" to describe
diffusion which is, of course, a large ka. approximation. For
ka <<l, we expect the loss to fall off. There may be a hint of

reduced scatter at 630 Hz which would make the scale size a = 0.5 mi.

Further support to the scattering hypothesis was given by the
experiments in the South Pacific (Bannister, 1976) which show an

excess absorption of 0.002 dB/kyd (see Figure 11). Like the North

398

ATTENUATION COEFFICIENT (dB/kyd)

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

T = THORP

HUDSON BAY
T+.04 dB/kyd

GULF OF ADEN
-6T+.02 dB/kyd

0.2 0.5 l Z 5 10
FREQUENCY (kHz)

Figure 9. SOUND-CHANNEL ATTENUATION (EXPERIMENTAL)

ATTENUATION COEFFICIENT (dB/kyd)

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

, 14 Y ouy 5 {

a) 1 2 5
FREQUENCY (kHz)

Figure 10. ATTENUATION COEFFICIENT IN
LAKE SUPERIOR

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ATTENUATION COEFFICIENT (dB/kyd)

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IN THE SOUTH PACIFIC

401

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

Central Atlantic, the North Central Pacific waters show no measurable

scatter. The Thorp coefficient is estimated to be 0.5 in both cases.

Another experiment, shown in Figure 12, was done much earlier in
the Gulf of Maine* and shows excess absorption similar to that for
Hudson Bay. In fact, the two are almost identical except for the
lower frequency points which are lacking in Hudson Bay. The dif-
fraction curve was based on infinitely lossy bottom as before and,
while the attenuation increases with decreasing frequency, the rate
is slower than predicted, probably because of finite bottom loss.

Any fall-off of diffusion loss at lower frequencies is obscured by

diffraction, however.

The latest experiment was done in Baffin Bay (Browning et al.,
1974) in 1974. The results in Figure 13 also show a constant loss
of 0.02 dB/kyd above 200 Hz with a rapid fall-off below that fre-
quency. Since Baffin Bay is much deeper than Hudson Bay or the
Gulf of Maine, the low frequency values are not obscured by dif-
fraction. The Be dependence below 200 Hz suggests Rayleigh scatter
from globs of scale size ay = 3 m with a = 2 x 10’.

From ray diffusion theory we have devised a simple formula,
shown in Figure 14, for the sound-channel diffusion attenuation in-
volving ah the variance of index of refraction, aor the scale size,
and Az, the depth from the channel axis to the bottom. Using the
values W? = 10° and an = 15 m obtained from analysis of the SVP in
Hudson Bay and the Mediterranean, we see that the values predicted
for shallow channels (Az = 100 m) and the deep channels (Az = 2,000 m)

are in reasonably good agreement with experimental values. If a_ is

* Unpublished BBN data.

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SOUND PROPAGATION IN A RANDOM MEDIUM

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i
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205

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DIFFUSION ATTENUATION

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MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

changed to 3 m, as predicted from Rayleigh scatter in Baffin Bay,
the result is 0.03 dB/km and agrees very well. It thus appears

that while most of the variability of scattering loss depends on the
strength of the channel, the smaller scale size is responsible for

the large value for Baffin Bay.

Although the values of scattering loss may or may not be impor-
tant to a sonar problem since it can be very small, the information
about what is happening to signal coherence within the channel cer-
tainly should be valuable. For example, we can see in Figure 15 the
effect of scatter on 400 Hz signal fluctuation for two hydrophones
separated by 100 m. In this experiment done by Stanford (1974) in
Bermuda, the time fluctuations are quite incoherent and seem to have
two scales, the longer one probably related to internal waves and a

shorter scale that may be related to turbulence.

The spectrum of the time fluctuations (Figure 16) definitely
shows a break above 10 cycles/hour which varies with the seasonal
thermocline. The latter scale size compares to that associated with

scatter loss if the ocean currents are one- or two-tenths of a knot.

The effect of scatter on spatial coherence is also important.
Kennedy (1969) at Bermuda varied the vertical separation of two
hydrophones and measured the CW signal correlation between them.
The correlation distance appears to be close to our estimated value

based on as = 15 m (Figure 17).

SUMMARY

Our results are summarized in Figure 18; we have observed both
the Thorp (boron) relaxation and also what we believe to be forward
scatter loss in a number of sound channels throughout the world. The

coefficient of the Thorp term is unity in the North Atlantic water

406

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(0) a ee mars al eo | hea esis L zt: [4 l ee ee L

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FOR TWO HYDROPHONES SEPARATED
BY 100 METERS

407

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

—— MARCH & SEPTEMBER
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but only 0.6 in the Gulf of Aden and 0.5 in the Pacific, suggesting
possible differences in the boron chemistry of these waters. The
regions where large amounts of scattering occur suggest turbulence
induced by current shear. In the North Central Pacific and North
Central Atlantic and Red Sea, there is no measurable scatter at all.
(The point of division between moderate and severe scatter is taken
as Ufa, = Mena rather than the magnitude of scatter loss since

the latter depends strongly on the channel strength.)

REFERENCES

Bannister, R. W., "Discontinuities observed in low-frequency long-
range propagation," in these Proceedings, 1976.

Browning, D. G., "Project CANUS: Sound propagation and reverberation
Measurements in Hudson Bay," NUSC Tech. Report 4221, 1 Dec. 1971.

, "Environmental factors affecting low-frequency propagation,"
in these Proceedings, 1976.

Browning, D. G., R. H. Mellen, J. M. Ross and H. M. Merklinger,
"Low-frequency sound attenuation in Baffin Bay," 88th ASA Meeting,
St. Louis, MO.., 5-8 Nov.. 1974.

Browning, D. G., E. N. Jones and W. H. Thorp, "Low-frequency attenu-
ation in the Gulf of Aden," NUSC Tech. Report 4501, 5 Mar. 1973.

Browning, D. G., and W. H. Thorp, "Attenuation of low-frequency sound
in the ocean — NUSC Research Program 1967-1972," NUSC Tech. Report
458), 4 Jane lO72:

Browning, D. G:, W. H. Thorp, F. C. Friedel and Rak: LaPlante,
"Project Hiawatha long-range shallow-water sound propagation in

Lake Superior," USL Tech. Memo. No. 221-173-68, 20 June 1968.

Chernov, L. A., "Wave Propagation in a Random Medium," New York:
McGraw-Hill, Chapter II, 1962.

DiNapoli, F. R., "Fast field program for multilayered media," NUSC
Tech. Report 4103, 26 Aug. 1971.

410

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

Kennedy, R. M., "Phase and amplitude fluctuations in propagating
through a layered ocean," J. Acoust. Soc. Am., 46:737-745, 1969.

Mellen, R. H., D. G. Browning and J. M. Ross, "Attenuation in randomly
inhomogeneous sound channels," J. Acoust. Soc. Am., 56:80-82, 1974.

Stanford, G. E., "Low-frequency fluctuations of a CW signal in the
ocean," J. Acoust. Soc. Am., 55:968-977, 1974.

Thorp, W. H., "Deep-ocean sound attenuation in the sub- and low-
kilocycle-per-second region," J. Acoust. Soc. Am., 38:648-654,
LOGS

Yaeger, E., F. H. Fisher, J. Miceli and R. Bressel, "Origin of the

low-frequency sound absorption in sea water," J. Acoust. Soc. Am.,
53 1e/O5—Ik/ Od O7S'

411

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

DISCUSSION

Dr. D. C. Stickler (APL, Pennsylvania State University) :
Referring to Figure 17, what was the difference between the correlation

that fell off rapidly and those that didn't?

Dr. Mellen: Two hydrophones at Bermuda were separated in the
vertical. The correlation between the two of them was measured as
a function of the separation distance. The dashed curves are from

Chernoff estimating at 100-feet and 200-feet correlation distance.

Dr. P. W. Smith (Bolt, Beranek, & Newman, Inc.): You picked out
my favorite example of something I completely fail to understand.
They have here a single path going up which may be significant on a
ray picture vertexing 124 feet, I think it was, below the surface,
then going down to the bottom, coming in at very shallow grazing
angle. And what I completely fail to understand is how they can get
such high correlation in the arrivals -- in the phase or arrival
times -- over their separation between the transducer pairs and this
very short correlation interval in the amplitudes. Does anyone have

any guesses?

Dr. Mellen: I can't answer that question. I only used this to
illustrate the correlation distances to compare with the 15 meters

that we measure in the Mediterranean and the Hudson Bay.

Dr. Smith: I don't think that the behavior of the time and
amplitude would be consistent with the theory with which it is being

compared.

Dr. Walter H. Munk (Institute of Geophysics and Planetary Physics,
Univ. of Calif., San Diego): There is something else I completely
don't understand, and other non-acousticians who have looked at your
results are equally confused. Figure 16 has power spectral densities

in dB, and I don't understand those units. Spectral density is units

412

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

of something per unit frequency band.

Dr. Ira Dyer (Massachusetts Institute of Technology) :

That's mislabeled actually.

Dr. R. R. Goodman (Naval Research Laboratory): On your last
picture (Figure 18) where you showed the areas of anomalously high
absorption, with the exception of the Gulf of Aden, they are all

shallow water results, aren't they?
Dr. Mellen: Baffin Bay is also deep water.
Dr. Goodman: How deep is it?
Dr. Mellen: About 2,000 meters.

Dr. Goodman: One thing I would like to point out with respect
to the Hudson Bay results and perhaps Lake Superior as well as any
shallow water. You are putting a tremendous amount of faith in the
shallow-water propagation loss that you are taking out of these data.
If you are talking about shallow-water propagation over a hundred miles
you are talking about an accuracy out to a hundred miles of 5 decibels
in the model, and that's better than any model I know today for

shallow water.

Dr. Mellen: Again, all we do is linear regression to the data.
We don't worry about absolute values or how it got there. We start
at very long distances. For instance, in Baffin Bay we measure only
from 100 kilometers to 400 kilometers in that region. And the
propagation is extremely well behaved, and you really believe the

results.

Dr. Goodman: You're subtracting off a transmission loss term.

You have to be.
Dr. Mellen: Subtracting out cyclindrical spreading.

Dr. Goodman: Right. Do you have faith in cyclindrical spread-

413

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

ing to that accuracy?
Dr. Mellen: Absolutely.
Dr. Goodman: That's curious, because I don't.

Mr. P. H. Lindop (Admiralty Research Laboratory): We have some
unpublished results for sound channels in the Western Mediterranean,
Eastern North Atlantic, and the Southern Norwegian Sea. Looking at
these rough results we don't see anything anomalous. We take out
cylindrical spreading and we don't see anything of the order that

you have seen.
Dr. Mellen: In the Mediterranean?

Mr. Lindop: In the Western Mediterranean and the Eastern North

Atlantic.

Dr. Mellen: I'm not quite sure where these results came from.
These were taken from Bill Thorp's notes, and were part of the JOAST
experiment. Two areas, one in the Tyrrhenian Sea and the other east of
Malta, were both measured. Now, if this was east of Malta, there is a
very strong ocean front which could be responsible for the relatively

large amounts of scatter that were observed.

Mr. Charles W. Spofford (Office of Naval Research): In the Hudson

Bay and Baffin Bay, were you using 1/3-octave filtering?

Dr. Mellen: Yes. All the experiments are 1/3-octave filters.

We haven't progressed to the sophistication of FFT.

Mr. Spofford: Have you taken your mode model or FFP and run it
to simulate the 1/3-octave filter to convince yourself that the spread-

ing is cyclindrical when viewed through the 1/3-octave filters?
Dr. Mellen: Dave Browning is going to talk about that tomorrow.

Dr. F. D. Tappert (Courant Institute of Mathematical Sciences,

New York University): Your shallow-water results have been criticized,

414

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

and I'd like to take issue with the deep-water results. It's very
difficult over long ranges to accurately compute the transmission loss,

and it's known not to be simple cyclindrical spreading in general.
Dr. Mellen: You're talking about the PARKA results now?

Dr. Tappert: PARKA and the South Pacific results. And you

showed some results for the Atlantic, deep water.

Dr. Mellen: Right. That was Thorp's original data set for the
North Atlantic. The North Central Atlantic didn't show any scattering.
In Thorp's original compilation, there is no scattering at all except

maybe a very, very tiny bit at the extremely low frequencies.

But on the ATOE experiment there was strong evidence of scatter-
ing. We can do two experiments. In one experiment we see lots of

scattering, and in the other experiment we don't. And it's real.

Dr. Tappert: I'm sure the effect is real. But whether we can
measure it quantitatively and make agreements with theory is another

issue.

Dr. Mellen: Well, let's say using this technique we found the
boron relaxation -- which nobody believed at that time. So now we
are finding something else besides the boron relaxation. We're talk-

ing about finding scatter.

Dr. Smith: I have a comment which stems from our treatment of
some of the Gulf of Maine shallow water data. I thought I'd try some
curve-matching to the data. The water was roughly 200 meters deep.
We had transmission loss data in third-octave bands at ranges from
something like 2 kilometers to 150. I thought I'd match the curve
for 1 kHz with the equation

i CN logan 1OuR

for values of N equal to 10, 15, and 20, choosing C and @ in each case

415

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

to get a best match. Obviously, I was going to select either 10, 15 or
20 depending on the standard deviation of the data points from the
trend curve. However, the standard deviation for those three values

of N ranged only from 1.1 to 1.4 dB. I decided that was not a very
sensitive test. There was a more sensitive test: The match for N=20

had a negative value of Qa!

If one is going to use 10 log R + QR as a transmission loss law,
you'd better be sure that all significant components of the energy are
being attenuated at the same rate, because that is a fundamental
assumption behind the law. A good example where it fails is the
classical shallow-water, isospeed theory where the modal attenuated
coefficient is quadratic in the mode number; this leads to transmission
loss varying as 15 log R+ QR. This example illustrates the fact that
the slope of transmission loss curve with range is strongly affected

by differential attenuation of the different components.

In our massaging of the data in the Gulf of Maine at 1 kHz we
found that this kind of differentially attenuated energy was at least
as important as the ducted energy out to ranges of something like 40
kyd. That was the transition range where they were about equally

important.

Now, the trouble is, if you start out at 40 kyd in order to be
sure that most of the energy is ducted and not differentially attenua-
ted, you've got only a factor of four on range before you start
running into noise or land as the case might be. You can't make a
very sensitive test of scope with the available data given the normal
scatter of points from any trend curve. Based on my personal experi-
ence I would be very skeptical about assuming 10 log R in cases

such as that.

Dr. Mellen: We have an initial spherical spreading region

while the sound channel is being set up. After that we have a mode

416

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

stripping region through which we progress into a region of one mode

propagation.

For example, in Baffin Bay, the smallest range was 100 kilometers,
the largest one was 400. The sound channel filled up in mode strip
much before that. So we had no problems there from say 50 kilometers

on to 400.

Dr. M. Schulkin (Naval Oceanographic Office): I don't consider
it using the same attenuation behavior if you have to take a half of
Thorp or six-tenths of Thorp or 0.75 of Thorp. You're not really

tying things down.

Dr. Mellen: No, of course not, but these are things now that
can be examined. If it turns out that the Thorp coefficient is con-
stant in the Pacific and the Gulf of Aden and so forth, the same as
it is in the North Atlantic, then we are going to have a check that

there is something wrong with our experiments.

Right now I say that for some reason or other there is less boron
absorption in the Pacific than there is in the Atlantic. I don't

know why.
Dr. Schulkin: It's a hypothesis.
Dr. Mellen: Yes.

Dr. Schulkin: But let me check one more point. The parabolic
equation requires 5 log R. Is this true? The intensity varies as

1 over R?

Dr. Tappert: No, there is cyclindrical spreading built in but
on top of that you have all other attenuation mechanisms that may

exist.

Dr. Weinberg: In all of these experiments the source and the

receiver were very carefully placed in a well defined channel and

417

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

were already beyond the region where the bottom is important. What

other mechanism is there if there isn't cylindrical spreading?

Dr. Smith: Well, I agree with you if you accept conventional
wisdom that the bottom strips everything off and there are no contri-
butions left and if then you accept that the ocean is homogeneous
you're going to get cylindrical spreading. There are situations

where I know some of those assumptions are not right.

Dr. Weinberg: Right, but we are not talking about those
situations. We are talking about very carefully planned experiments
where we are sure to put things right on the axis or as close as

possible.

Dr. Smith: Gulf of Maine may not be one of those you want to

point at then.

Dr. Tappert: One problem. The very theory that explains this
scattering attenuation predicts that the fluctuations will also fill
in modes as they are stripped off. As some are stripped off, others
are filled in by the random fluctuations. So you are not left with
just the single mode. Therefore, the attenuation without the

scattering will not be purely cylindrical.

Dr. Goodman: We are talking about a very tiny effect ona
large propagation, and what we really have to do is determine the
confidence limits we have on what is left over. I think the only way
this will ever become convincing to anyone is to have a very careful
analysis of all of the elements that contribute to the total loss and

some sort of error analysis on how well you can trust your models.

Most of us don't have that kind of faith in our models. I think
it's up to you to put down some numbers so that the confidence limits
are valid for taking a guess. We would certainly like to see something

like boron. It's an interesting problem. But the question really

418

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

is: have we seen evidence that is statistically significant after

we have subtracted off all of these things?
Dr. Mellen: You still don't believe the boron?
Dr. Goodman: I haven't seen any error analysis.

Dr. Smith: I want to suggest in that same vein that it is very
difficult to establish confidence. We need all the measurements we
can get, not just total integrated transmission loss. An experiment
should be planned to make other measurements of transmission, whether
it be signal envelope, coherence, directionality, or whatever. By
using that information as well as the transmission loss and testing
the results against a model for the physics of the situation, one's

confidence would be increased.

Mr. R. L. Martin (New London Laboratory, Naval Underwater
Systems Center): I believe that the original work that Bill Thorp
did at least prior to 1968 did not use total energy at all. He looked
at the peak envelope of the classic SOFAR arrival, and he was just
concerned with the amplitude of that envelope and how that changed

with range.

He did not use the total energy, but he did make a comparison
of the two methods in the PARKA exercise and they came out with the

same results.

I just bring up that point to indicate he initially dealt only

with those rays that were very close to the axis.

Dr. smien: Some of his work I thought he time-gated and then
got the energy in what he thought was ducted.

Mr. Martin: Not his initial work though.

Dr. Mellen: All the later stuff was total energy in the window.

419

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

Mr. Martin: When he did that work in the PARKA exercise he did

find he didn't get any difference.

Mr. Smith: Of course, those are two experiments which are parti-
cularly well designed in nice deep water, good channels and good
Measurement positions. Also, we want to remember the recent work
that was interpreted as showing negative bottom reflection losses.

We have got to bring all our information about the physics together,

it seems to me, and make a consistent whole of it.

Mr. Pedersen: There is a problem with using peaks of convergence
zones to do this because if you include the diffraction correction,

the loss drops off as ee

Mr. Martin: But it isn't a convergence zone. It's the SOFAR
shape. While there is a lot of rays in there that are adding coher-
ently and somewhat incoherently, you're doing regression analysis to

get rid of the incoherent.

Mr. Pedersen: If you look at one caustic in the convergenze
zone and you identify it in the first one and second one, and so on,

that level drops off as mele

Dr. Hersey: I think I'll make one attempt to bring the wrath
of everybody down on my head because this particular controversy has
been bubbling in our community for some time and it has seemed to me
that as we have talked about it and as new results have become avail-
able from various parts of the world, some very imaginative choices
have been made of experimental locations, and there is an excellent
body of data available from just the group at New London that has
done so much of this work, and there are other data samples like the
PARKA set that they participated in, all of which are sufficiently
well documented to see what the attenuation coefficient is as a

function of the data analysis model.

420

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

We now have the parabolic equation model. We have some rather

sophisticated ray models.

It sounds to me offhand as though it might be feasible to make
that kind of a study and see whether the attenuation coefficient re-
sults are highly dependent on the model used to reduce the data. Be-

cause the data are bound to be good data.

Dr. Mellen: We rely mostly on the agreement with the FFP, al-
though I don't know whether we have used CONGRATS 5 or anything like

that yet.

Dr. Hersey: That, you see, you have right in your own group.
But here's a case where we have an excellent data bank. The fact that
the work was concentrated in one place is perhaps a strong argument
for consistency in the way the work was done. And if we as a
community would manage to make use of our several model designs to see
how sensitive the reduction of the data is to models, we would at
least have a basis for answering some of the worries that were expressed

here this afternoon.

When that has been done we would all have a basis for making an
estimate of what we ought to do next. Is it a critical experiment?
Or a critical series of experiments? Or are we beginning to approach

understanding?

Dr. S. M. Flatte (University of California, College of Santa Cruz):
I wanted to mention that last summer our group made some parabolic
equation runs which attempted to indicate what the scattering in the
sound channel would be from internal wave models. I would emphasize
that you have to have a model of everything that is happening in the

ocean to decide what the scattering is.

However, it would be at this point rather easy for us to propagate
any number of modes through internal waves and find out what the

scattering due to that would be.

421

MELLEN: SOUND PROPAGATION IN A RANDOM MEDIUM

Dr. Mellen: From your data, presented last night, I made an

estimate it would be not measurable.
Dr. Flatte: You mean because of the range?

Dr. Mellen: No, because of the large scale size of the internal
waves. In other words, I think we're dealing with a smaller scale size

which increases the scattering loss.

Dr. Flatte: If you really have to go down 15 meters, then you're

right. Internal waves will not explain it.

Dr. Mellen: It isn't that far off though, because even though
your scale size for the horizontal is much larger, your vertical
scale size is smaller because of the ellipticity in the internal

wave inhomogeneity.

Dr. Flatte: The vertical scale size is what I'm talking about.

The vertical scale size in the internal waves is like 200 meters.

Dr. Mellen: The scattering will depend upon the diffusion
constant which goes as Wa, (see Figure 14) in the geometrical
acoustics limit. But if these inhomogeneities are not spherical as
we said, then they are multiplied by the ratio of the horizontal major
axis to the vertical minor axis. You get that much more diffusion if

the things are lenticular.

Dr. Flatte: Then we're going to multiply your 15 meters by a
factor of approximately 10 to account for the ellipticity, which

makes it very close to internal waves.

Dr. Mellen: Right. Yes.

422

PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

R. C. Spindel

Woods Hole Oceanographic Institute
Woods Hole, Massachusetts

Observations of low-frequency, long-range acoustic
transmissions have revealed a correspondence between
acoustic phase variations and internal oceanic effects
such as tidal cycles, transport phenomena, geostrophic
flow and internal gravity waves. For periods less than
the local inertial period and greater than the local
bouyancy (Brunt-Vaisala) period, internal waves appear
to be the predominant cause of acoustic phase fluctuations
(in the absence of severe multipath). Measurements of
220 Hz and 406 Hz transmissions at ranges from 200 to
1200 km using free-drifting receivers of varying depth
have substantiated this conjecture. The empirical acoustic
phase spectrum is proportional to a theoretical phase
spectrum constructed by using a simple ray theory in con-
junction with a hypothesized internal wave spectrum (Garrett
and Munk). Furthermore, a predicted dependence of fluctu-
ation energy on depth is observed in these data.

These measurements have been used to determine a
mixed space-time coherence function as a function of range
which establishes the oceanic limit of array resolution.
The simple ray-internal wave theory predicts coherence
parameters that compare favorably with data. Data collected
to date have suggested several important areas for future
consideration.

INTRODUCTION

Recent observations of low-frequency, long-range acoustic trans-
missions have revealed a correspondence between acoustic phase vari-
ations and internal oceanic effects such as tidal cycles, transport
phenomena, geostrophic flow and internal Rossby and gravity waves
(Steinberg, et al., 1973; Weinberg, et al., 1974; DeFarrari, 1974;
Baer and Jacobson, 1974; Franchi and Jacobson, 1973; Spindel, et al.,

1974; Porter, et al., 1976; and Stanford, 1974). From a physical

423

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

oceanographic standpoint the causal relationships between ocean param-
eter variability and acoustic phase variability suggest techniques for
measuring oceanographic phenomena. From an operational viewpoint, the
performance of detecting and tracking systems is strongly dependent

on the phase stability of the ocean transmission path. In both cases

the connection between ocean parameters and acoustic phase must be

understood.

Fixed system studies (in which source and receiver are rigidly
attached to the ocean floor) conducted in the 200 to 800 Hz region
of the spectrum have shown that for periods less than the local
inertial period and greater than the local bouyancy period, internal
gravity waves appear to be the predominant cause of acoustic phase
fluctuations. These periods range from about 5 minutes to 1 day at
a latitude of 30°. Time scales of this order are of utmost interest

in array tracking and detection applications.

PHASE FLUCTUATIONS

Phase fluctuation data collected at Woods Hole exhibit most of
the features found in data obtained by the Institute for Acoustic
Research in Miami, the New London Laboratory of the Naval Underwater
System Center, the Bell Telephone Laboratories, and others. The
experiments conducted at Woods Hole, however, have significant
differences, and this is reflected in some of the observations we
have made. Woods Hole data are not obtained with a fixed system.
The acoustic source is moored at varying depth, and receivers are
either free-drifting, towed, or moored. Receivers are suspended in
mid-ocean at depths varying from 300 to 1500 meters. Receiving
hydrophones sweep out synthetic spatial and temporal apertures

several kilometers in length and several hours in duration.

424

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

The technique used to form these apertures and to compensate
them for motion of the receiving hydrophone has been reported in a
previous paper (Porter, Spindel and Jaffee, 1973). In brief, receiving
hydrophones are suspended within range of a bottom moored navigation
net consisting of three acoustic sources emitting continuous tones in
the 12 to 13 kHz region. Receiver motion is manifest as Doppler shifts
in these tones. Doppler shifts are translated into equivalent motion.
With the current version of the system, receiver motion is tracked to
within 1/4 wavelength at 12 kHz, about 3 centimeters. Long-range
acoustic transmissions at 220 and 406 Hz are simultaneously received
by the moving hydrophone. Doppler shifts due to receiver motion are
resolved into equivalent phase shift at 220 and 406 Hz. This shift
is subtracted from total accumulated phase leaving a residual phase
variation in the long-range transmission resulting solely from

variations in the intervening water mass.

Figure 1 shows the deployment of a typical navigation net and the
generation of five distinct apertures labelled 130, 131, 132, 133, 135
from 3 to 8 km in length. The time span of each aperture is indicated

by time in minutes along each drift path.

Figure 2 is a schematic illustration of received low-frequency
transmission. The carrier at 220 or 406 Hz is received at frequency
te displaced from - by the Doppler shift due to receiver drift.
Spread about f. results from variation in drift rate, and from vari-
ations in the transmission medium. When scattering from the sea
surface is significant, it appears as sideband energy about the
carrier with peaks at multiples of the peak frequency of the surface
wave spectrum. The Doppler correction scheme removes variations fa:

The signal is then heterodyned down to dc, and variations resulting

from surface scatter are removed by filtering around the carrier.

425

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Figure 1. TYPICAL NAVIGATION NET DEPLOYMENT (CIRCLES)
AND SYNTHETIC APERTURE GENERATION

426

COHERENCE AND INTERNAL WAVES

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SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

The result of this processing is a signal at dc, the fluctuations of

which are due to fluctuations in the ocean transmission path.

Figure 3 illustrates the ray path geometry between the point of
signal transmission and a receiver located about 210 km away. Typical
samples of received acoustic phase are shown in Figure 4 for a receiver
at a depth of 300 meters. Over an approximate 3-hour interval spanning
an aperture of 3.5 km, peak-to-peak phase fluctuations are about 7
cycles. Two more examples of raw phase fluctuations are shown in
Figures 5 and 6. Here we have compared fluctuations at two frequencies
approximately an octave apart. Both frequencies were recorded and
processed simultaneously. Careful examination of these figures
indicates that observed phase fluctuations are approximately twice
as great in the 406 Hz data. This suggests that the scale of
inhomogeneities encountered by the acoustic transmission is large
compared to a wavelength. Thus, the transmissions are affected

independently, and notions of simple frequency scaling appear to hold.

One implication is that large-scale phenomena, internal waves
for example, are primarily responsible for fluctuations in this

frequency range.

INTERNAL WAVES AND PHASE FLUCTUATIONS

Some rather simple theoretical ideas contribute strongly to our
assumption that internal waves are the predominant factor in generating
phase fluctuations at these acoustic frequencies and ranges. The
frequency of internal wave oscillation is bounded at the lower end
by the local inertial frequency and at the upper end by the local
bouyancy frequency, n(z), a function of depth. Figure 7 illustrates
the relationship between sound velocity variations and internal wave

parameters. Sound velocity fluctuations are proportional to the

428

COHERENCE AND INTERNAL WAVES

PHASE FLUCTUATIONS,

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Depth = 305 m
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406 Hz

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431

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Depth = 305 m
Range = 200 km

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432

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INTERNAL WAVE OSCILLATIONS ARE
BOUNDED BY BUOYANCY FREQUENCY n(z),

n@(z) 6a

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TO LOCAL BUOYANCY FREQUENCY

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

amplitude of the internal wave 6 and the temperature gradient ee
which in turn is proportional to the bouyancy frequency. Thus,
maximum variations in sound speed due to internal wave action occur
at the depth where n(z) is greatest. This is usually near the main
thermocline as illustrated by the sample bouyancy frequency profile

shown.

On the basis of this analysis, the internal wave field is modeled
as affecting a traveling acoustic wave only in a thin layer at thermo-
cline depth as shown in Figure 8. A ray passing through the layer
will experience a phase advancement or retardation depending upon
whether the immediate sound velocity of the layer is greater or less
than the average sound velocity. Figures 9 and 10 outline the
theoretical analysis necessary to complete the internal wave-acoustic
wave interaction model. The internal wave field is modeled as a random
superposition of waves concentrated in a layer of thickness n. The
field is characterized by a frequency-wavenumber spectrum proportional

to the internal wave model proposed by Garrett and Munk (1972).

The phase change A@ of an acoustic signal due to a single passage
through the internal wave layer is proportional to acoustic frequency,
the angle with which the ray enters the layer at and the internal wave
spectrum. The spectrum of the resulting acoustic phase variations
Eve, is proportional to the number of times the ray has passed through
the layer, M, and the square of the acoustic frequency. It is also
a function of the inertial frequency Ws and cuts off at the local

bouyancy frequency no:

Figure 11 shows a plot of this theoretical spectrum as a heavy
solid line together with measured phase spectra for receivers at two
different depths. The light solid line represents data at 1500 m,

the dashed line at 305 m. Transmission range was about 200 km. These

434

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data fall off with a slope of approximately of -2 as predicted by the
model, and in this respect these data lend credence to the model. Data
taken at greater depth show more rms phase fluctuation which supports
the notion of an equivalent internal wave layer. Rays to the deeper
phone have spent a greater fraction of their travel time in or about
the layer. There is no cut-off at the bouyancy frequency, contrary

to the model prediction. This result has appeared in the work of
others, Stanford (1974) for example. At present we attribute this

lack of abrupt fall-off to the contaminating effects of microstructure
which may begin to dominate at higher frequencies. We shall return

to this point below.

It seems safe at this juncture in our current understanding of
phase fluctuations to assert that internal waves are the dominant
cause of fluctuations at these acoustic frequencies and that such
fluctuations range in period from several minutes to a day. It is
important to appreciate that the term "internal waves" can cover a
host of phenomena, including tidal waves, Rossby waves, more classic

internal waves, and wavelike behavior of microstructure.

COHERENCE

Oceanic induced phase fluctuations establish limits on array
performance. Upper bounds on coherent array processing gains are only
approached when the signal received across the array is phase coherent
from array element to element. The pointing accuracy or resolving
power of an array is critically dependent on the phase coherence of
the acoustic transmission path. Figure 12 illustrates these ideas.

A simple two-element array of length L receiving energy from a distant
acoustic source (point source) is said to be working at the utmost
limit of its resolving power when the random phase difference along

the two paths is less than 1/2 cycle. This phase difference is

439

COHERENCE AND INTERNAL WAVES

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SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

equivalent to a path length variation of A/2 meters. The beamwidth of
the array at this limit is A/2L radians. Fluctuations in phase re-
sulting from the ocean environment must therefore be less than 1/2
cycle or A/2 meters for the array to achieve its diffraction limit.

If the phase fluctuations are greater than 1/2 cycle, the resolving

power of the array is said to be environmentally limited.

The acoustic-internal wave theory outlined above predicts an
rms path length change that is proportional to f and to the number of
times the ray crosses the internal wave layer, i.e., distance. It
predicts that rms phase fluctuations will reach a limit at some
separation of sensors, and that the magnitude of fluctuation at this
separation is proportional to distance. It is interesting to note that
the performance of an environmentally limited array continues to
increase linearly with array length since R, = P/L, and P becomes

constant.

Figure 13 shows phase fluctuation data at 406 Hz and two ranges,
200 and 1200 km, as a function of array length. Both curves rise to
a plateau, about 13 meters of equivalent rms path length change at
200 km and 40 meters at 1200 km. Theory predicts values of about
15 and 45 meters, respectively. The environmental limit at 200 km
would thus be avoided for all A/2 > 13m, or f£f < 50 Hz. At 1200 km,
f < 20 Hz ensures diffraction rather than environmental limited per-
formance. These curves were computed from data gathered during
synthetic aperture formation and therefore represent a limit imposed
by temporal as well as spatial variations. In that sense, they can
serve as an upper bound on coherent array performance. It is expected
that actual performance of a fixed spatial array will be somewhat

better.

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CONCLUSIONS, PROBLEMS, AND RECOMMENDATIONS

This paper has presented some illustrative data and theoretical
notions that summarize much of our current appreciation for the phase
fluctuation problem in long-range acoustic transmissions. We have
restricted our presentation to Woods Hole data because we feel that
they illustrate the major effects of internal waves on phase stability,
and its consequences regarding array performance. Some large data
sets obtained by other researchers support our notions, while others
give us pause. Longer time series illustrate effects not seen in our
data, such as tidal cycles and variations resulting from seasonal
changes. These are important, too, and critically so if acoustics
is to be used as a tool for studying large-scale oceanographic

phenomena.

An example of the type of behavior we do not fully understand
is shown in Figures 14A and 14B.* It shows the amplitude and phase
spectra of a 367 Hz tone transmitted between Eleuthera and Bermuda.
The phase spectrum falls off as eo with no apparent cut-off at the
local buoyancy frequency. The amplitude spectrum, however, falls
rapidly at the buoyancy frequency. Our feeling has been that environ-
mental effects would be most visible in the acoustic phase, and that
multipath effects would so distort the amplitude fluctuations as to
make environmental-acoustic amplitude comparisons difficult indeed.
Apparently this is not the case for data such as these have been
obtained by the Institute for Acoustical Research and others. A
similar spectrum of amplitude fluctuations calculated at Woods Hole
using transmissions from free-drifting SOFAR floats at 270 Hz anda

range of 600 miles is shown in Figure 15.** Again the buoyancy

* Reproduced from a paper by G. Stanford (1974).

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PERIOD, HRS
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FROM SOFAR FLOAT

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

frequency cut-off is quite clear and dramatic. This is only one illus-
tration of our present lack of understanding and consequent inability

to predict and forecast.

A firm connection between oceanographic effects and acoustic
effects must be established to allow the most effective exploitation
of ocean transmission paths. Well controlled experiments are necessary
to sort out the host of contributing factors such as microstructure,
internal waves, cyclonic and anti-cyclonic eddies. Experiments must
be conducted in a variety of locations to learn whether results at
one point can be extrapolated to another. Similarly, experiments
must be conducted at a variety of frequencies and ranges to establish
the scaling laws so necessary for accurate prediction. Perhaps most
important of all, however, is the need to coordinate acoustic experi-
ments with strong physical oceanographic programs, so we can signifi-

cantly increase our understanding of acoustic variability.

REFERENCES

Baer, R. N., and M. J. Jacobson, "Analysis of the Effect of a Rossby
Wave on Sound Velocity in the Ocean," J. Acoust. Soc. Am. 55,
1178-1189, 1974.

DeFerrari, H. A., "Effects of Horizontally Varying Internal Wave
Fields on Multipath Interference for Propagation Through the
Deep Sound Channel," J. Acoust. Soc. Am. 56, 40-46, 1974.

Franchi, E. R., and M. J. Jacobson, "An Environinental-Acoustics Model
for Sound Propagation in a Geostrophic Flow," J. Acoust. Soc. Am.
5S eos —S4 7), 97sec

Garrett, C., and W. Munk, "Space Time Scales of Internal Waves,"
Geophys. Fluid Dynam. 2, 225-264, 1972.

Porter, R. P., R. C. Spindel, and R. J. Jaffee, "CW Beacon System

for Hydrophone Motion Determination," J. Acoust. Soc. Am. 53
LOGUE OS; SeltO7 Se

447

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Porter, R. P., R. C. Spindel, and R. J. Jaffee, "Acoustic-Internal Wave
Interaction at Long Ranges in the Ocean," J. Acoust. Soc. Am. (in
press).

Spindel, R. C., R. J. Porter, and R. J. Jaffee, “Long Range Sound
Fluctuations with Drifting Hydrophones," J. Acoust. Soc. Am. 56,
440-446, 1974.

Stanford, G., "Low Frequency Fluctuations of a CW Signal in the Ocean,"
vg. Acoust&. Soc. Am. 55, 968-977, 1974.

Steinberg, J. C., et al., "Transmission Fluctuations," Institute for
Acoustical Research, Miami Division, Palisades Geophysical
Institute, Final Report IAR 73001, June 1, 1973.

Weinberg, N. L., J. G. Clark, and R. P. Flanagan, "Internal Tidal
Influence on Deep-Ocean Acoustic-Ray Propagation," J. Acoust.
Soc. Am. 56. 447-458, 1974.

DISCUSSION

Dr. S. M. Flatte (University of California College at Santa Cruz):
I think there is a difficulty in the way you treated the effect of
internal waves on the phase fluctuations. Before I describe the
difficulty, let me say that I believe that treating it properly
will not change your qualitative result, with which I agree whole-
heartedly — that is, that internal waves cause the type of fluctua-
tions we are observing. But I think it will change the quantitative

comparisons.

If you take a source and a receiver which are connected by a path
such as shown in Figure 8, then the question is: What is the region

of this path where the internal waves make the biggest effect?

You suggested that there was a fixed depth. The fluctuation
: : : , 2
formula which you gave (in Figure 10) for Foe. varied as l/sinw@

where re is the angle the ray makes with respect to the internal wave

448

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

layer. You immediately see that there is a problem when We Se Ol.

That is, Fae? blows up.

Dr. R. C. Spindel: We are quite aware of that.

Dr. Flatte: Right. I am sure you are. The result of this, of
course, is if this layer happens to occur at the horizontal turning
point of the ray, then this does not apply any more because the ray
is actually curved. The point is, though, that the path does spend a
great deal more time in the layer near its turning point than in any

other layer that it is traversing.
Dr. ‘Spindel: Yes.

Dr. Flatte: From our studies, at least in the type of profile
we were working with, which was quite different from considering a
particular layer, a factor-of-10 more time is spent in the region

near the upper turning point than in any other region.

So I would be surprised if your profile was such that the effect
at the turning point, which has a factor-of-10 enhancement due to the
flatness of the ray, was unimportant compared to the region of your

fixed depth.

Dr. Spindel: Yes. We're quite aware of the limitations of the

ray theory, and that is basically —

Dr. Flatte: This is not a limitation of the ray theory. That
is, I think you could apply the ray theory with this except that the
result would be you would get most of your contribution from the

region where the ray is flat.

Dr. Spindel: Yes, if that is the region where the internal waves
have their largest effect. I think they do in that portion of the

water column.

449

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. blatte: That ais sight.

Dr. Spindel: That is true. We are aware of that. We are also
aware of the sensitivity of the model to selection of that angle, 6
Even if you were to assume that that layer were not in a particularly
difficult area (where the rays turn, for example) but were lower, the
model is quite sensitive. Angles are quite shallow at which rays

enter and leave the layer.

We are not really propounding the theory as one which explains
all the interactions between internal waves and acoustics. But what
we wanted to point out was that the environmental effects of internal
waves are mirrored in the acoustic phase. And I think we can do that
although we cannot predict absolute levels, which is basically what

that factor is.

Dr. Flatte: I agree with you completely and that, in fact,
the results do show the internal waves compare quite favorably with
these data. I would like to make one more comment that has to do
with the one I made last night about the difficulty with computer

codes.

Roger Dashen and Walter Munk did an integral over the ray path
and found that in fact for our case the main contribution came from

the turning point.

The theoretical prediction which was given to me to compare with
what came out of the computer code was that if you plot the rms phase
fluctuation, as a function of range, you expect rather small fluctu-
ations before the turning point. And as soon as you reach the area
of the turning point, there should be a rather sharp jump. When I
looked at the computer code, I did see a reasonably small fluctuation

up until the turning point, although the quantitative agreement has

450

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

yet to be established. At the turning point I also observed a sharp
rise, but I can't tell how much because it was above a cycle. And
at that point we are stuck, whereas your data are fine enough that
you can follow the phase through several cycles and find out what is

happening.

Dr. R. P. Porter (Woods Hold Oceanographic Institute): I want
to reinforce your comment. We attribute the depth dependence we see
in our phase fluctuations precisely to that turning point argument,
coupled with the fact that the layer of nearly constant sound
velocity occurs right near the region where the internal wave

activity appears to be the greatest.

We feel it is a qualitative conclusion that we really can't test
accurately because of the breakdown of the ray theory in that region.

But we have come to that same conclusion.

Dr. Flatte: Why do you think the ray theory has broken down?
Why not just integrate the true path through that region? You know
the length.

Dr. Porter: Because it is a caustic. I do not think it is

valid —

Dr. H. A. DeFerrari (Rosenstiel School of Marine and Atmospheric
Science, University of Miami): We do not know where the ray turns.
There is an ambiguity. As the grazing angle becomes small, any
slight perturbation to the sound-speed profile causes a turn. So
if you want to integrate through there, you may not be on the same

ray that strikes your receiver.

Dr. Porter: Put it another way. In that ray you have diffrac-

tion effects.

451

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Flatte: The integral converges. There are certain regions
where you have to worry about it breaking up and causing a completely
new ray. But you can also form criteria for that not happening. We

have done some work in that respect, too.

Dr. DeFerrari: This is a problem that was in the Acoustical
Society several years back. Whether or not one added in a phase
shift of “1/4 at that point or not. This 1s pant of that integral
you are talking about. I don't know whether it was ever really

settled or not.

Dr. P. W. Smith (Bolt, Beranek, and Newman, Inc.): Yes, it was

NOE. (Laughter )

Dr. Flatte: I have another question concerning the quantitative
comparison that you made of the phase fluctuations. How did you
treat the combination of several rays? There were several rays going

from source to receiver. Right?

Dr. Spindel: Four rays.

Dr. Flatte: How do you treat the combination in order to get a

total phase prediction for the model?

Dr. Spindel: The total field at the receiver is simply a
summation of the effects of those four rays. We have computed the
phase at the receiving point for each of the four rays, we sum that,
and separate that resulting equation into an amplitude and a phase

factor and that is the phase.

Dr. Flatte: So the internal wave model predicts the phase
fluctuation of each ray and to compare with data you perform a

summation of those rays?

452

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Spindel: Yes.

Dr. Flatte: But in a random way?

Dr. Spindel: The randomness comes in because of the randomness
of that layer. That layer is a superposition really of many internal

waves.

Dr. Flatte: But the internal wave predicts the average fluctu-
ation in one ray or another. It does not predict how the combination
will occur unless you assume you know the amplitude of fluctuations
and their distribution and then form some kind of a statistical

combination.

Dr. J. G. Clark (Institute for Acoustical Research): You did

not describe internal wave field statistics?

Dr. Spindel: Yes, and the resulting received signal is really

just a superposition, that is, a linear combination of all the rays.

Dr. Flatte: You assumed equal amplitude?

Dr. Spindel: Yes.

Mr. C. W. Spofford (AESD, Office of Naval Research): I have
here three figures that are the results of a numerical experiment
which I think bears on these phase statistics. It was stimulated
by a question I asked Bob Porter about a year or so ago at an
Acoustical Society meeting when I first heard of the technique of the
drifting floats, because I was concerned that he was taking out the
phase assuming that it was essentially linear in range. And I think

we have seen ample evidence today it is not.

I actually made a numerical calculation using the parabolic-

equation program extracting the phase as a function of range at

453

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

selected depths. Wewent out one convergence zone, using a frequency
of only 50 Hz because such computations are too expensive at 400 Hz.
A Bermuda-type profile was used over a high-loss bottom with a source
on the axis. Figure D-l shows the transmission loss and phase
variations as a function of range. The phase has been de-meaned

over the 50-mile interval and the residual variations are plotted

in cycles. Note that in this case of axis~to-axis propagation,

the phase varies from linear only by about one quarter of a cycle.
But here is the propagation loss going along. I don't know if I

am willing to multiply it by 8 to scale it up to 400 Hz or not.

One of the phase flip questions occurred to me when examining
the results near 22 nautical miles. There was a particularly deep
null in transmission loss, and the phase changes by nearly 180

degrees; actually, it is about 135 degrees.

We ran this case again with very fine resolution in range and
the phase was totally continuous through there. There were no
discontinuities. I concede that if you are measuring the phase near
such a point, and the signal level has dipped down into noise there
is no way to track the phase. But there is no reason, no physical

reason, for the phase to be discontinuous.

Figure D-2 corresponds to the same source but to receiver depth
of 300 feet. Here the phase and loss curves overlap. Note that the
phase-variation scale has been compressed to handle the 14-cycle

variation over the entire range.

As you come up into the convergence zone, some fairly dramatic

things are happening in terms of phase.

Figure D-3 is for a 1000-foot receiver depth where the up-and-
down-going convergence zones overlap more. Here the phase scale is

changed again.

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SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

The point I wanted to raise is that phase, even in a frozen
deterministic ocean, is not linear in range. And I am concerned with
techniques which remove a linear phase trend and assume that the
residual fluctuations are due to internal waves or other random-

ocean effects.

In summary, there is a natural non-linearity in phase in the

ocean.

Mr. M. A. Pedersen (Naval Undersea Center): You took out a

constant velocity, didn't you?
Mr. Spofford: I accumulated a phase for the entire range.

Mr. Pedersen: Yes, but you removed it by taking out a constant

phase velocity?

Mr. Spofford: Yes. Essentially.

Mr. Pedersen: And as you move into different parts of the con-
vergence zones you have different phase velocities because you have

different vertical angles.
Mr. Spofford: Absolutely.
Mr. Pedersen: So it will progress this way.

Mr. Spofford: I was expecting to get phase variations. The
question I was not sure of was how quickly they might change with

range.

Now, his measurement only went over a fraction of a mile, I think,
in range. So I do not think it is driving the problem in terms of the
kinds of things he measured. I think it will introduce a variation

or non-linearity in phase on the order of about one cycle at 400 Hz.

458

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Clark: Over what?

Mr. Spofford: Over the fraction of a mile. There is a ripple

on the order of a cycle in phase.

Dr. Clark: You took a linear phase trail out of this?

Mr. Spofford: I went back and looked at half-mile regions,
removing the linear trend, and still ended up with residual fluctu-

ations of about a cycle in phase extrapolated to 400 Hz.

Dr. Clark: You made a comment on that last slide that you did

not see any reason for a phase flip in the deep fade. Is that right?

Mr. Spofford: No, I did not say that. I said the phase flip
is continuous in the model at least. Such physics does not have
discontinuous phase. The problem is, it is always at these nulls in

transmission where you are probably looking at noise.

Dr. Clark: Right.

Dr. M. Schulkin (Naval Oceanographic Office): One remark on
the last slide that Dr. DeFerrarri showed. Were those Doppler shifts

measured?

Dr. DeFerrari: No.

Dr. Schulkin: Was it just coincidence that you chose 8 seconds?

Dr. DeFerrari: I started at 8 and then on down.

Dr. Schulkin: They are as close to sinusoidal effects as you
are going to get in the ocean. You are apt to pick up an 8-second

swell by measurement, and I just wondered if you did.

459

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. DeFerrari: No, the first one was measured and it probably

was 8 seconds. They are typically 8 to 10 seconds.
Dr. Schulkin: It was measured?

Dr. DeFerrari: Yes. The first set of a few. Yes. ‘That iis
what it looks like. The swell comes in very strong. The wind-driven
waves are much less compared to the real spectrum. I have done that
with surface data at the same time and compared the spectra and it
looks just about like the wave spectra, that the wind waves fall

out much more rapidly than the actual spectrum.

I have also done it as a function of frequency and a number of

other things.

Dr. R. M. Fitzgerald (Naval Research Laboratory): I wanted to
make a quick comment on the physical nature of discontinuous phase
jumps. What we have is a physical field, a pressure field. When
you decompose that field into phase and amplitude, that is unphysical,

if you like.

However, when the amplitude vanishes, the phase is not deter-
mined. So when the amplitude truly vanishes, the phase can change

discontinuously in the physical pressure field.

Dr. T. G. Birdsall (Cooley Electronics Laboratory): Some people
have had a lot of experience trying to read data through those
points, because it is the nastiest point in the world. The nicest
thing is to run three frequencies through it, you know, an epsilon

apart on either side —

Mr. Spofford: I would argue in a deterministic physical model

like this that the amplitude probably cannot vanish.

460

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Flatte: The chances of it vanishing are zero.

Dr. Birdsall: But it happens, though. That's the trouble with

probability zero. It always keeps happening.

Dr. S. W. Marshall (Naval Research Laboratory): This question
is to Bob Spindel. Bob, you made a statement that beyond the limit
that the environment places on the array you can expect to get gain

from that array. Please clarify that.

Dr. Spindel: You can continue to get increased resolution by
increasing the size of your array. This is simply a consequence of
the fact that as you separate sensors, the phase fluctuations between
the two sensors appear to saturate at a particular level. They do

not increase beyond that level.

So your angular resolution is determined by that phase fluctu-
ation divided by the length of the array. So that you can do better
and better by making your array longer and longer. It does not mean
that you should do that. You might be buying very little. Asa
matter of fact, you do buy very little every time you double a long

array in terms of the expense of doing it.

Dr. W. H. Munk (University of California at San Diego): May
I make two comments? One, to those of us who are pushing internal
waves as a cause of acoustic fluctuations, it certainly is dis-
concerting, to say the least, that acoustic spectra seem to pay no
attention to the high frequency cutoff of internal waves. Spectra

merrily go by without change in slope. I don't like it.

But I do want to point out something kind of interesting. The
same happened to be the case for measurements of the up and down

motion of the internal waves. All of such measurements before

461

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

January 1974 did not show any effect of the Vaisala frequency, and
all measurements since January 1974 have shown sharp cutoffs. Very

curious discontinuity.

There at least we do know the answer other than people finding
what is fashionable at the moment. Those before were moored measure-
ments, and the whole field was convected past the fixed transducers,
so that to some extent you got a mixture of the spatial as well as
the temporal variation. And there is no cutoff in the spatial vari-

ation at high wave number.

So you might think of that as a Doppler shift or a Doppler smear
cutoff. If the tides convect your whole field by variable speed, it

is certainly going to blur and maybe even eliminate the cutoff.

When people went to other kinds of instrumentations, like
capsules that yo-yoed but stayed with the water column, then all of
a sudden the cutoff did, thank heaven, appear. It was a 20 dB cutoff

and was very pretty.

And I am hoping against hope somehow that in some future experi-
ments in acoustics suddenly a sharp Vaisala cutoff will appear. I

do not know how.

Dr. Clark: Where would you expect that, Walter?

Dr. Munk: At the local Vaisala — I don't know. That is a good
point. There is, of course, great smearing if you have rays which
have gone through the whole water column. I hadn't thought of that.
And that really in a way changes the situation from the internal

wave experiment I mentioned.

Dr. Clark: In that data that I gave you, you will find a knee

at about one cycle per hour.

462

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Munk: That is a very good point, John. For those acoustic
measurements which sample a large depth of water column you would,

in fact, have a good excuse for not seeing a sharp cutoff.

Dr. Clark: It is not sharp but there is a definite knee there

so you can see that.

Dr. Munk: The other point is more serious. I have an impression
that these phase and intensity fluctuations are very model-dependent
that no matter what kind of a model you put into the ocean, as long
as it is not complete nonsense, you are going to find excellent agree-

ment with observations.

I have seen this now at least in four or five different models.
Jacobsen puts in planetary waves, and by cooking them up a little
they show a record that he says looks like precisely the record

that you obtained.

Other people like us have put in internal wave observations and

they look lovely. So does your group.

And, finally, you, to make it even worse, show that under certain
conditions you don't need any disturbance at all. You just need

interference.

I think there is a lesson to be learned there one snould face —
that when it comes to multipath, the statistics you get are probably
highly dependent on path interference and very unsensitive to the
Ocean model itself which is good if you are an acoustician, because

you might get some good results without having to study the ocean.

It is bad for us oceanographers because we are probably not going
to be able to use that kind of statistics ever to learn anything

sensible about the ocean.

463

SPINDEL: PHASE FLUCTUATIONS, COHERENCE AND INTERNAL WAVES

Dr. Clark: Can I make a comment, Walter? On that first model
study I did in the Straits of Florida, I tried to predict some
statistical characteristics of the amplitude rather than the phase.
This is a more significant comparison, I believe. As we know,
amplitude is a non-linear function of the environment. So, if you
can predict the frequency content of that thing, then you have done

something.

Let's hope if we go ahead and do the complete job, that every-

thing will smear out.

Dr. Birdsall: I hope the ability of all the models to predict
it means that we do have to get more quantitative and perhaps richer
experiments where we measure more than just one kind of thing so

that we can start to split across the various types of models.

464

FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING
MULTIPATH AND DOPPLER SPREADING

H. A. DeFerrari

Rosenstiel School of Marine and Atmospheric Science

Signals transmitted through ocean channels will be spread in
time because of multipath and spread in frequency because of
scatter from the ocean wave surface. Fixed-system measure-
ments in the Florida Straits and between Eleuthera and Bermuda
make possible the observation of time-varying multipath inter-
ference and Doppler spectra. Results are summarized for
several short experiments using CW (420 Hz) and pulse CW trans-
missions.

Fully coherent ray models are used to interpret experimental
results. These models predict the transmission loss and travel
time along all paths with sufficient accuracy to allow the co-
herent addition of arrivals at the receiver. Time-varying CW
multipath interference is simulated by introducing perturbations
to the sound-speed field and generating time series of phase
and transmission-loss fluctuations for comparisons with experi-
mental results. Model computations show that horizontally in-
variant internal waves produce sound-speed perturbations that
cannot cause both the phase and transmission loss fluctuations
which are consistent with experiment. When horizontal fluctua-
tions are introduced to the sound-speed perturbations, statis-
tics of CW transmission fluctuations match experimental results.

Pulsed CW transmission can also be simulated by coherent addi-
tion of received pulses. Broadband characteristics of received
signals exhibit selective fading. The frequency of the fade

is sensitive to small perturbations of sound speed while the
fade bandwidth depends on average characteristics of the pro-
pagation channel and is relatively insensitive to the typically
observed fluctuations of sound speed.

Doppler spectra and scattering functions are presented and dis-
cussed. Combined propagation and scattering models show that
unsymmetric surface-scatter sidebands can result from bottom
interactions.

465

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH
AND DOPPLER SPREADING

This paper consists of the presentation and. discussion of some re-
sults from fixed-system measurements. Basically three types of pheno-
mena are described:

e Time-varying multipath for CW signals
e Time-varying multipath for broadband signals
e Doppler spreading

Before presenting these data I would like to discuss a model used
for their interpretation. The basic model is a bilinear profile with
quasi-static fluctuations. A surface scattering model is included on

each path, and the paths are added coherently.

Figure 1 shows typical range-averaged profile between Eleuthera
and Bermuda and a bilinear fit to it. We don't have any experimental
data of the sound speed fluctuations there, so for perturbations of this
profile we use a calculation made by Dr. Moore at the University of
Miami of the first-mode internal wave for an internal tide of wavelength
150 kilometers. Figure 2 shows the resulting perturbed profiles, and
Figure 3 shows the bilinear approximations to them. The perturbations
can then be described in terms of two parameters: the depth D of the
axis and the bilinear angle a. This model can be mode range-dependent,
as shown in Figure 4. The sound-speed profile becomes a function of
range by segmenting it and allowing the profile to change with range
and also with time. Figure 5 shows the bilinear fit to some actual
sound-speed measurements made about mid-range in the Florida Straits.
One profile was obtained every two hours for four days. If you look at
the sequence closely, you can see the effects of a tide. The gradients

change and the knee rises and falls by the tidal periodicity.

The bottom sketch in Figure 6 is the surface scatter model we will
use. There is a specular reflection, unshifted in frequency, from the
surface waves, and Doppler-shifted sidebands separated in frequency

from the carrier by multiples of the surface-wave frequency. We have

466

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND
DOPPLER SPREADING

SOUND SPEED (m/sec)
Peoe 1500 1520 1540

DEPTH (km)

Figure 1. TYPICAL RANGE-AVERAGED PROFILE AND BILINEAR FIT

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND
DOPPLER SPREADING

SOUND SPEED (m/sec)
1480 1500 1520 1540

PTH (km)

oe
O

Figure 2. PERTURBED PROFILES FOR FIRST MODE INTERNAL WAVE

468

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND
DOPPLER SPREADING

SOUND SPEED (m/sec)

l480 I500 [S20 I540
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469

FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND

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TYPES OF RAY PATHS

SOURCE 47 NN REC E

SRBR

SCATTERED SRBR

C=Acos (kx - wet

\
v ‘N
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"ECKART" SCATTER MODEL
Figure 6. RAY PATHS AND SURFACE SCATTER MODEL

472

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH
AND DOPPLER SPREADING

algorithms that can track rays incident on the surface all along the
propagation path and find those scattered rays which hit the receiver.

Thus, multiple scattering along the path can be accounted for.

The types of ray paths that can be expected in the Florida Straits
are shown in the other diagrams on Figure 6. The downward refracting
profile leads to rays which are refracted bottom-reflected (RBR) and
surface-reflected bottom-reflected (SRBR). The SRBR rays can be
surface scattered (as in the illustration) after any number of specular
bounces. This happens to be a ray which reflects specularly twice,
then upscatters and reflects specularly as its new angle before reach-

ing the receiver. The model does all the bookkeeping for these paths.

We will now discuss the data. Figure 7 shows CW propagation loss
for the 700-mile range between Eleuthera and Bermuda over a 48-hour
period; typical multipath deep fades (30 dB or so) are shown with their
associated phase shifts. If the fades are very deep the phase shift
appear to be 180 degrees. Most of the phase fluctuations are quasi-

periodic, varying with the tidal component.

It's interesting to compare these sorts of fluctuations with what
we see at other ranges. Figure 8 compares propagation-loss and phase
data for the 700 nautical mile range to Bermuda, the 300-mile range
to Eleuthera and the 7-mile range in the Florida Straits. They all
have the characteristic dropouts in signal level due to multipath
influence. The principal difference between them is that for the
longer ranges the fades are more rapid than for the shorter ranges.
However, the fades tend to have the same magnitude, typically 15 to
30 dB for deep-fading events. The phase has similar characteristics,
with smooth variations (showing a strong tidal periodicity in the
Eleuthera data) plus a number of rapid shifts of 180 degrees associated

with deep fades.

473

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH AND

DOPPLER SPREADING

32
TIME IN HOURS

C.W. TRANSMISSION BETWEEN ELEUTHERA AND BERMUDA (407Hz)

Te)

100
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474

Figure 7.

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BERMUDA
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-140!

= fa ae

ELEUTHERA 3
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PHASE (CYCLES)

FLORIDA STRAITS (7NM.)

TRANSMISSION LOSS (dB/nb)

-10 aa el (ae a. ta

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TIME IN HOURS

Figure 8. TRANSMISSION LOSS AND PHASE FOR THREE RANGES

475

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF TIME-VARYING MULTIPATH
AND DOPPLER SPREADING

Histograms for each of the above time series are shown in Figure 9.
They are all about the same with a standard deviation of about 5.5 GB.
The essential difference in the three cases is the autocorrelation
function. In Figure 10 it is seen that the Bermuda series decorrelates
more rapidly than the other two. It appears more noiselike. If we
take the decorrelation time to be that value at which the normalized
autocorrelation function falls to 1/e, then we get about 4-1/2 minutes

for the Bermuda range, 8-1/2 for Eleuthera and 18 in the Florida Straits.

Another measure is the mean square bandwidth (defined at the top
of Figure 11) of the power spectrum of the transmission-loss time
series. Figure 11 shows that the more noiselike Bermuda time series

has a broader bandwidth.

Now, the characteristics that we have tooked at so far are ones
which are really consistent from day to day over long periods. But we
must be able to differentiate between the fast fading events and the
intermediate ones. Figure 12 is a sequence of histograms for a time
series of 63 days. Each time series is high pass filtered with a
cutoff of 4 cycles/day so the periods of variation are less than 6
hours. All the longer periodicities are removed. Note the spectra
day after day are consistent and formally speaking appear to be
wide-sense stationary. Figure 13 shows the corresponding autocorrela-
tion functions, again for 63 days. Again these are consistent one

after another.

In the above figures, the longer term trends were filtered out.
Figure 14 shows variations for periods longer than 5 hours which are
significant and I don't think are related to multipath. I can't think
of any mechanism for them offhand other than it may just be a complete

change in the whole propagation regime. It appears that there are

476

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different paths so we may be looking at, say, a twentieth convergence
zone Sliding back and forth relative to the receiver. There are also
significant variations {20-dB or so) on the order of a day which have
been omitted for the time being from the study. I don't think they are

multipath either.

Figure 15 shows the Bermuda phase record again. The phase has
long-term trends in it as well as the tidal component. I have chosen
to differentiate this record to look at the rate of change of phase as
another variable for comparison. Figure 16 shows these results. Long-
term trends have very slow rates of change so they don't contribute
very much. One thing that stands out is the large tidal component for
both the Eleuthera and the Bermuda ranges. They have about the same
average rate of change of phase 6, and they have about three or four
cycles of change per tidal period. This appears to be a good place to

start on some model comparisons.

The first thing that we do with the bilinear profile (shown at

the top of Figure 17) is to take a perturbation which is constant

with range but varied in time. That is, let the whole profile rise and
fall like the first-mode internal tide. The next thing is to adjust

the amplitude of the fluctuation so that it gives the right amount of
phase shift. However, when we do that, we don't get enough interference.
The fades don't come as frequently as they do in the experiment. In
fact, there doesn't seem to be any way that you can adjust this profile
in this manner to get anything else but the kind of variations shown.

The fluctuations that yield large shifts in phase don't give enough
amplitude interference, and they decorrelate in about a half hour in-
stead of the four minutes typical of the experiment. This also seems

to be true for fluctuations which have a scale larger than the cycle dis-

tances of the SOFAR rays. We have tried using an internal tide starting

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on the shore and dissipating while propagating seaward. The effect was
the same as taking the perturbations and distributing them evenly over
the whole range. Basically it provides enough phase shift but not

enough interference.

However, using the tidal frequencies and adding a little bit more
perturbation to the system we can get the same amount of interference
that we see in the experiments. Figure 18 illustrates the procedure
where a 4-meter random internal-wave component is added to the decay-
ing tide. It adds no significant contribution to the phase (see middle
graph) other than the same jitter. However, the jitter introduces
more fades (bottom graph). Each ray basically interacts with fluctua-
tion components of comparable cycle distances. Hence they select the

appropriate component from the internal-wave spectrum.

There are a lot of other modes that could be added to the pertur-
bation but all I have put in are tidal-like frequencies. I have broken

it up spatially, and it seems to be enough.

The fades that we see are not strictly continuous wave -- that
is they have some bandwidth associated with them. One way to measure
it is to transmit a broadband signal, a pulse and look at the received
time series. Figure 19 illustrates the result of transmitting a 20
millisecond pulse. The signal that arrived was about 100 milliseconds
wide representing the superposition of many pulses with slightly
different travel times corresponding to different RBR rays. Alsoa
lower level group is seen which appears from the model studies to be

an SRBR arrival.

Figure 20 illustrates the behavior of such pulses during the time

that a CW signal is fading. The top figure shows the CW amplitude

487

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AND DOPPLER SPREADING

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which exhibits a deep fade. The phase of the signal (shown below)

goes through a 180-degree phase jump at the fade. For the pulse (bottom
figure) a small notch forms on the right side of the pulse which

slides across the pulse with increasing time. Precisely in the middle
of the CW fade the pulse power is also very low. Traveling with the

notch is a 180-degree phase jump.

What this says about the CW fade is that at the instant of the
deepest part of the fade, the energy is equally split into two components
which are 180 degrees out of phase with each other and hence cancel.

On the other side of the fade the resultant vector shows up 180 degrees
reversed from before the fade. The only way this can happen is if

the perturbation that's causing it is causing all the arrivals --

there are 15 arrivals in the pulse -- to slide relative to each other.
So it appears to be a broad-scale process rather than a localized

fluctuation.

Figure 21 shows a model simulation where the gradient shifts slow-
ly with time. The pulse response is in the left column and the phase
is in the right colum. A small notch forms in the pulse and slides
across the pulse, notching it out. Traveling with the notch is a
180-degree phase shift. These results contain 15 arrivals each with

slightly different travel times.

An alternate representation of this fading is shown in Figure 22
in terms of a series of measured power spectra of successive pulses.
The carrier is the center line at 420 Hz. What happens here in time
is that we are going through a CW fade. Transmission is falling off
and coming back up again on the carrier. For the full spectrum it is
apparent that the fade slides across the band resulting in selective

fading.

491

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF THE TIME-VARYING MULTIPATH
AND DOPPLER SPREADING

MAGNITUDE

PHASE
____PHAS
ae

sate

2 MIN,/ FRAME
SIMULATED

Figure 21. MODEL SIMULATION OF PULSE EVOLUTION FOR
SLOWLY VARYING SOUND-SPEED GRADIENT

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POWER SPECTRA VARYING IN TIME

Figure 22.

DEFERRARI: FIXED-SYSTEM MEASUREMENTS OF THE TIME-VARYING MULTIPATH
AND DOPPLER SPREADING

Figure 23 illustrates a modeled case for a typical sound-speed
profile measured in the Florida Straits with a small perturbation added
to it. On the left is an amplitude-frequency-time plot and on the
right is a phase-frequency-time plot. Note the two fades at early
times. As time goes on they slide across the band. Traveling with
the deep fade is a 180-degree fade shift. The other fade isn't quite

as deep and its phase shift is somewhat less than 180 degrees.

While no one would claim to be able to predict when these fades
will occur, certain features are predictable, notably the bandwidth.
The bandwidth depends on average characteristics, not on the detailed
fluctuations in the sound-speed profile. The precise time of the fade
is determined by extremely small changes in the profile and hence is

not predictable.

The models not only predict the frequency response but can also
simulate spatial processing; for example, a coherent summation at several
points. The modeled fade cells as shown in Figure 24 are small and
isolated at 100 Hz. One of the few advantages of ray theory is once
you make this computation you can change the frequency and easily
consider several frequencies. Figure 25 is the same kind of plot for
200 Hz. (Note: there's a scaling of a hundred to one from range to
depth so these contours are actually very elongated. The contour in-

terval is 5 dB.) Figure 26 is the same thing at 420 Hz.

I would now like to present some Doppler-spread data. Figure 27
is a typical Doppler spectrum I measured in the Florida Straits. The
carrier line at 420 Hz has been suppressed to emphasize the sidebands.
The sidebands are characteristically asymmetric and differ by 3 to 6 GB
almost always. The spectrum appears to be a replica of the surface-

wave spectrum.

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One of the things I wanted to do in the model work was to include
all possible scattered arrivals as well as the specular paths and to
look at the resulting distribution for a pulse both in time and fre-
quency. Figure 28 shows the modeled results for levels as a function
of time. The top line represents the RBR rays with their characteris-
tic buildup as they stay closer and closer to the bottom. Successive
arrivals have one additional bounce. SRBR arrivals (second line) tend
to spread out because they are traveling up and down and each order has
a significantly greater travel time. They have a little spreading-loss
anomaly in the beginning, and then drop in amplitude as a result of

the surface interactions.

The model predicts that the Doppler-shifted energy is going to
come in and peak out somewhere behind the main RBR group. The first
SRBR doesn't have any arrivals that get there at about the same time.
The later ones have one or two. Then they peak out with four or five.

The surface bounces then start to take over.

Figure 29 is a measurement of this process. The Doppler spectrum
has been measured for each successive part of the received signal for
a transmitted pulse. Repetitive pulses are actually used to obtain
these data. They come in just the way the model says at about the

right intensity.

These computations gave me enough confidence in the model to
attempt the deep ocean case. Figure 30 corresponds to the 700-nautical-
mile case. The bottom line shows the refracted-refracted (RR) rays
coming in with various intensities. The top two lines show the RBR
rays and SRBR arrivals. The third and fourth lines show the up- and
down-Doppler scattered arrivals, respectively. Our model keeps track

of all these arrivals, and the Doppler spectrum is predicted to be

asymmetric.

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I would like to go back now and look at one specular arrival and
the family of scattered arrivals that have the same number of
surface interactions. Figure 31 illustrates the SRBR structure plotted
in terms of the intensity versus arrival time for the same total
number of surface reflections (25). The number next to each arrival
indicates the number of specular bounces it had made prior to emitting
the scattered ray which happens to hit the receiver. The one marked
"24" made 24 specular bounces before it emitted the ray. Note that
the rays which scatter at the ends (either near the source or near the

receiver) suffer the least loss.

The same information can be expressed in terms of the grazing
angle that the arrivals make with the bottom after scattering (Figure 32).
All the arrivals that interact in the last half of the received pulse
have a significantly lower grazing angle, about 5 to 10 degrees, than
all those that are in the first half. Also the first half are all
upscattered, whereas the second half are all downscattered. The
difference in the bottom loss with these different grazing angles is

enough to cause the consistent 3 to 6 dB sideband asymmetry.

Figure 33 is the predicted Doppler spectrum (or more properly the
transfer function which must be multiplied by the surface-wave spectrum
to get the Doppler spectrum). Note the 3 dB difference in the side
bands. The scattering event itself is symmetric. Because of the
differences caused by: (1) the angle at which the ray is emitted from
the surface and (2) the requirement that the ray hit the receiver,
the upscattered paths have significantly less loss than the down-
scattered paths. These results are quite consistent with what is

observed in measured data.

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DESIGN OF TRANSMISSION LOSS EXPERIMENTS

J. S. Hanna

Office of Naval Research
Code AESD

The conceptual design of transmission loss measurements is
discussed. The concern here is not with the hardware
implementation of a desired measurement, but with the
definition of what is to be measured given everything we
know about the medium, the acoustic sources, and the
available processing techniques.

The cyclical effort of the past in which models were used
to interpret data, and the data in turn used to refine
models, is drawn upon to illustrate some general proper-
ties of the impulse response of the ocean. Given these
general properties, the following topics are addressed:

1) The nature of sound sources (impulsive and continuous
wave) and the limitations each imposes upon our
ability to measure the spectrum of the ocean's
impulse response

2) The selection of a signal processing scheme (analog
or digital), given the expected nature of the impulse
response and the properties of the sound sources

3) Examples of measurements which, in some cases have
and in others have not permitted meaningful interpre-
tation of the results; these examples illustrate
common problems and the way they can be avoided.

INTRODUCTION

The objective of any transmission loss experiment is to measure
a particular property of the ocean environment, namely its effect on
the transmission of an arbitrary signal between two points. This
seemingly obvious statement is worth making because some measurements
have been conducted in a way which has inextricably confused the
properties of our measurement system (source and signal processor)

with those of the medium.

509

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

If one could determine the impulse response of the ocean between
any two points, then, in principle, one can predict what will happen
to any signal which propagates between these two points. Asa
practical matter, however, one can only aspire to measure a band-
limited version of this impulse response. If it is furthermore
realized that the modelers are interested in not just an experimental
determination of the impulse response, but in its interpretation
through physical properties of the environment, it becomes clear
that measurement planning must consider: 1) the expected properties
of the impulse response, and 2) the limitations imposed by signal
sources and processors upon the measurement of this impulse response.

In the course of this paper, both these topics will be considered.

THE IMPULSE RESPONSE

For the purpose of illustrating some properties of the impulse
response, it will be assumed that the medium is not dispersive
(that is, the medium simply attenuates the amplitude equally at all
frequencies and introduces at most a phase reversal upon reflection
from the ocean surface). Consider, then, the hypothetical, idealized
impulse response of Figure 1 which consists of four arrivals time-
delayed according to the history in the upper right-hand corner.
There are two pairs of arrivals separated by a time At. The total
history is assumed to correspond to the four arrivals of a single
order for some source-receiver geometry. For the sake of example
the two time differences and amplitudes were selected as shown and
the spectrum of the resulting impulse response displayed in the

figure.

By way of interpretation, the 40 Hz periodicity corresponds to
the time delay At while the 2 Hz periodicity corresponds to the time

delay AT. For a more complicated arrival structure there will be a

510

DESIGN OF TRANSMISSION LOSS EXPERIMENTS

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periodicity in the spectrum corresponding to each pairwise time delay
in the time history. (It will become apparent shortly that, depending
upon the desired characterization of the transmission loss, it is
necessary to anticipate the relative time history of arrivals and,
thereby, the general structure of the spectrum of the impulse
response.) Further elaboration of the model to include such features
as frequency dependent absorption and frequency .independent phase
shifts (such as produced by caustics) will cause details of the
spectrum to change; however, the basic periodicities induced by the
travel-time differences will remain. It is these periodicities which

will drive our later concerns.

SOME COMPLICATIONS

Limitations of Impulsive Sources

The desire to measure the spectrum of the impulse response be-
tween any two points runs rapidly into some practical difficulties.
To measure a spectrum, such as shown in Figure 1, requires a source
of energy with a flat, featureless spectrum over the frequency domain
of interest. In general, such sources can only be approximated,
often poorly. The most widely used impulsive source in Navy measure-
ment work is the explosive charge. However, because of the presence
of bubble pulses, these explosives themselves have a rich spectrum

which may rival that of the ocean's impulse response.

Examples of these spectra for 1.8-pound charges of TNT detonated
at 60 and 800 feet are shown in Figures 2 and 3, respectively. In
Figure 2 the rapid (6 Hz) variation is the bubble pulse frequency
while the slower (vV80 Hz) variation is caused by the surface-reflected
arrival. (Both spectra shown in Figures 2 and 3 are low-pass filtered

at 300 Hz.) It is quite possible to produce a 6 Hz period in the

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DESIGN OF TRANSMISSION LOSS EXPERIMENTS

100 150 200 250 300 350 400 450
FREQUENCY IN Hz

SPECTRUM OF THE DIRECT ARRIVAL AND SUR-
FACE REFLECTION FROM A 1.8-POUND CHARGE
DETONATED AT 60 FEET

500

RELATIVE SPECTRUM LEVEL IN dB

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DESIGN OF TRANSMISSION LOSS EXPERIMENTS

100 200 300 400 500
FREQUENCY IN Hz

FIGURE 3. SPECTRUM OF A 1.8-POUND CHARGE

DETONATED AT 800 FEET

600

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

spectrum of the impulse response discussed earlier, with an appro-
priate choice of receiver depth (about 800 feet in this case), which
would be indistinguishable from that caused by the bubble pulse of
the source at 60 feet. Similarly, referring now to Figure 3, the
source at 800 feet has a bubble pulse frequency of approximately

50 Hz (the surface reflected path has been gated out in the time
domain) and, again, a particular receiver depth could induce a com-

parable periodicity in the spectrum of the impulse response.

The first point to be made, then, regarding measurement of the
impulse response of the ocean is the nature of the limitation in-
duced by our attempt to produce a source with a flat spectrum. The
rule of thumb which follows from this point is that a source should
be chosen (or tailored) such that its distinct spectral features are
very different from those features of interest in the spectrum of

the impulse response.

A second potential difficulty may arise in the choice of a signal
processing scheme. If it is desired to measure the spectrum of the
impulse response in detail over a wide band, then a natural choice of
processing is digitization of the data and FFT spectrum analysis.
Even though this processing permits very narrowband analysis, some
frequency average of the spectrum will be desirable for at least one
of two reasons: 1) it may be necessary to average over spectral
variations of the source which are not strictly repeatable from event
to event (such as the 6 Hz variation in Figure 2), and 2) it may be
necessary to average over certain fine structure of the spectrum of
the impulse response itself which is known (or expected) to change
rapidly from measurement to measurement. These factors are further
explored by Hanna and Parkins (1974). This frequency average can be
selected only with knowledge of the detailed structure of both the

source and impulse response spectra.

515

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

For reasons of economy, or for the sake of real-time processing,
it may be desirable to do the necessary frequency average by band-
pass filtering the received signal before processing. The factors
affecting this choice are the same as those mentioned above. However,
the penalty for error is higher in this case: in the absence of

permanent broadband recordings, the measurement cannot be redone.

An example of an experiment in which the source and signal
processing choices proved well matched to the desired measurement
concerns the measured spectrum of an impulse response as shown in
Figure 4. The event was a 3-pound charge dropped at a range of
300 nm and detonated at a depth of 60 feet. At this range the total
received signal consisted of the arrivals from a single convergence
zone. The 5 to 6 Hz variation is caused by the bubble pulses of the
shot and the 220 Hz variation by the interference of the direct and
surface-reflected paths at the source. The received signals were
filtered through 1/3-octave filters at 25, 50 and 100 Hz; these
filters were wide enough to average out the bubble pulse effect, but

narrow enough to properly sample the surface image effect.

It is clear from Figure 4 that the received level will be about
10 dB lower (and, thus, the transmission loss will be 10 dB higher)
at 25 Hz than at 100 Hz. This expectation is borne out in Figure 5
which compares the measured transmission losses at these two fre-

quencies over the 500 nm range of the event.
Limitations of CE Sources

So far the discussion has been limited to the consideration of

broadband sources; these sources are well suited to measuring the

516

SPECTRUM LEVEL IN 48

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

-20

-30

-40

Figure 4.

200 300 400
FREQUENCY IN Hz

MEASURED SPECTRUM OF IMPULSE RE-
SPONSE FOR CONVERGENCE ZONE AT
300 NM

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400

300

200

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Figure 5.

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

impulse response spectrum for discrete source-receiver geometries.
There is much interest, however, in the behavior of a CW signal
radiated by a continuously moving source; the behavior is deter-
mined most directly by the use of towed CW projectors. In this

case, the spectrum of the impulse response is sampled at a single
frequency (with some narrowband resolution) as a function of the
changing source-receiver geometry. Even though the narrowband
sampling may be produced continuously in time (as, say, the output
of an analog filter), the practical question arises as to how often
{in time] should this output be sampled to give transmission loss

as a function of changing geometry. The answer is simple and obvious:
often enough to adequately represent the underlying continuous curve.
If, however, one must set up an automatic sampling system, it is
necessary to estimate in advance the character of the transmission
loss as a function of changing geometry, just as in the case of
impulsive sources it is necessary to estimate the spectrum of the

impulse response.

This point is illustrated in the next two figures. First, in
Figure 6 are shown an estimated transmission loss curve and its
experimentally determined counterpart for nearly axis-to-axis
propagation in the Mediterranean. The calculation was performed
using the parabolic equation program as implemented at the Acoustic
Environmental Support Detachment; this calculation was carried out
with a range resolution of 0.1 nm which was adequate to sample the
rapid variations of the loss with range. The data were taken at
approximately 5 nm intervals using a time average equivalent to a
range interval of 0.05 nm. Although the data are not inconsistent
with the calculation (and, thus, suggest that the real-world trans-
mission loss has character comparable to that of the estimated loss),

it is clear that the dependence of the actual transmission loss on

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range is grossly underdetermined. Second, consider the comparison of
Figure 7 which also presents both measured CW data and a parabolic
equation calculation. In this case the calculation was done with a
resolution of 150 feet in range while the data were taken with a

time average corresponding to a range interval of 30 feet. Although
different in detail, both curves have comparable structure; the
important point is not the level of agreement or disagreement between
the curves, but that both represent adequate spatial sampling of the
underlying transmission loss and that the model calculation could

have been used to set the experimental sampling intervals.

The chief advantage of a CW source is that it permits experi-
mental determination of the behavior of narrowband signals. A
significant disadvantage is that it seldom permits a path-by-path
analysis of the transmission loss. When properly used, the CW and
impulsive sources can provide information on complementary questions
regarding the nature of propagation. The impulsive source is suited
for measurements of the spectrum of the impulse response of the
medium for fixed source and receiver locations (with a frequency
average imposed by the nature of the source and, perhaps, even the
medium). The CW source is suited for measurements of the spectrum
of the impulse response at one frequency for continuously varying

source and receiver locations.

The Message

The central point in the above discussion is that proper design
of a transmission loss experiment demands a priori estimation of the
nature of the loss characteristic to be measured. The present state
of acoustic models, both ray and wave, certainly permits making these
estimates with high confidence in many cases. Historical precedent
is no longer a sufficient (or even necessary) reason for using any

signal source or processing technique.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS
A CASE HISTORY: THE CRITIQUE OF A BOTTOM LOSS MEASUREMENT

Introduction

The Naval Oceanographic Office (NOO) and the Naval Air Develop-
ment Center (NADC) have conducted many measurements over the past
several years aimed at determining the bottom reflection loss at
low frequencies (less than 1,000 Hz). Although different in detail,
the two programs have employed similar experimental techniques. They
both tend to use sources and receivers within some hundreds of feet
of the ocean surface; in this geometry transmission loss is measured,
compared with estimates of the spreading loss through the water, and
bottom loss is inferred. A consistent, and surprising, result of
most of those measurements is the apparent evidence of negative
losses at low frequencies for low grazing angles. This result has
serious implications for predicted transmission loss using present
models; in the remainder of this discussion the experimental design
employed in these measurements will be examined, along with its impact

upon the inferred reflectivity.

An Example

The case study here assumes a Pacific profile for the water
column and a sound velocity gradient in the upper few hundred feet
of bottom sediment of 1.0 ser (see Figure 8). This assumed velocity
structure for the unconsolidated sediment of the bottom is supported
by numerous independent experiments including those being considered
here. No discontinuity of the sound velocity into the bottom has
been assumed, although there is evidence that a discontinuity of a
few percent often exists. Its absence here is of no material conse-

quence for the points to be developed.

223

DEPTH (M)

HANNA:

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6000

7000
1460

DESIGN OF TRANSMISSION LOSS EXPERIMENTS

SEDIMENT

1500 1540 1580 1620
SOUND SPEED (M/SEC)
Figure 8. PACIFIC PROFILE

524

1660

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

The schematic ray trace of Figure 9 shows a typical source-
receiver geometry and the four ray-paths belonging to the family of
rays having one bottom interaction. Explosive charges are used as
sources and the experimental design assumes that these four paths
are isolated by the signal processing. The basic analysis uses the

following relationship applied to these paths:

RL = SL —- TL - BL
where RL = received level,

SL = source level,

TL = transmission loss,

BL = bottom loss.

The received level and source level are measured, the transmission
loss (excluding bottom loss) for the paths is estimated and bottom
loss is subsequently inferred. For the moment, it will be assumed
that there is no uncertainty in the measured received level or source
level (although the problem of source level measurements will be

touched upon later).

Transmission Loss Estimates
Bottom-refracted Paths

The examination here begins with the assumed transmission loss
model. In their data reduction, both organizations have assumed that
1) all four paths are of equal intensity at all ranges, and 2) the
contributions from all four paths combine on a power basis to yield
the total intensity. Based upon these assumptions, the total
spreading loss along the four paths is just 6 dB less than the loss
along any single path. Assumption 1) above is acceptable except for

ranges corresponding to small grazing angles on the bottom; at these

525

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Depth

FIGURE 9. SCHEMATIC RAY TRACE

526

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

ranges the paths begin to drop off in intensity at significantly dif-
ferent rates. Thus, the total loss estimated from a signle path will
be different at low grazing angles, depending upon which of the four

paths is used.

The second assumption above can lead to more serious difficulties than
the first. Based upon the earlier considerations of this paper, whether
any set of paths is combined in the signal processing on an rms basis
or not depends upon the details of the processing. In the experiments
of interest here the received signal was filtered in a 1/3-octave band
at several center frequencies. To assess whether assumption 2) is
reasonable, estimates of the spectrum of the impulse response for the
four paths of Figure 9 were made; these estimates were based upon the
computed amplitudes and arrival times for the paths. The relative
arrival times as a function of range are shown in Figure 10 for the
paths which refract through the sedimentary layer in the sound speed
profile shown earlier. (The minimum range corresponds to a path
incident upon the bottom at an angle of 20° with respect to the
horizontal.) Figure 11 shows the computed spectrum for a range of
14 nm; the 9 Hz variation is caused by the up-and-down-going pair of
paths at 800 feet, while the 27 Hz variation is caused by the up-and-
down-going pair of paths at 300 feet. At 35 Hz, for example, a 1/3-
octave filter is about 8 Hz wide at its 3 dB down points; a filter of
this width clearly will not yield the rms sum of the features of this
figure. Figure 12 shows the computed spectrum at a range of 29.5 nm;
all the travel time differences have decreased with corresponding
increases in the frequencies of the variations in the spectrum. Again,

the filter at 35 Hz will not yield the rms sum of these variations.

Figure 13 compares the transmission loss for these four paths
based upon 1) the rms sum, 2) a 1/3-octave result at 35 Hz, and

3) a 1/3-octave result at 100 Hz. In the portion of the figure above

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REFLECTIVITY LOSS (dB)

LOSS (dB)

ANGLE (DEG)

INFERRED REFLECTIVITY

RMS SUM
SHOT PROCESSOR 35 Hz
SHOT PROCESSOR 100 Hz

80
90
BOTTOM - BOUNCE
TRANSMISSION LOSS
100
(@) 10 20 30 40
RANGE (NM)

Figure 13. REFLECTIVITY AND TRANSMISSION
LOSS FOR STRUCTURE OF FIGURE 9

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

the transmission loss, the inferred reflectivity is plotted assuming
that the measured loss is either the 35 or 100 Hz filtered result
while the computed loss is the rms sum. There are two important
points to be made here: 1) the inaccurate transmission loss model
(viz., the rms sum) induces spurious character into the inferred
reflectivity which is not only frequency dependent, but source-
receiver geometry dependent; 2) the inferred reflectivity loss is

consistently negative for angles less than 6 to 7 degrees.

Water-refracted Paths

Up to this point is has been assumed that the four bottom-
interacting paths can be resolved from all other paths in the problem.
This is not always the case for ranges corresponding to low grazing
angles. To demonstrate this fact, consider first the ray plot of
Figure 14 where are shown the rays from a source at 800 feet which
arrive in the range-depth window from 25 to 35 nm and O to 300 meters.
Those paths which reflect from the surface are distinguished according
to whether they interact with the bottom or belong to the RSR family;
also shown are the RR rays. Consider the rays which intersect the
receiver depth at 300 feet: the last bottom-interacting path arrives
at a range of 31 nm, yet even the ray-trace shows non-bottom-
interacting paths arriving in the overlapping range from 29.5 to 31
nm. In reality, however, the refracting paths make their influence
felt before 29.5 nm in the form of the shadow zone field of the RR
caustic. The relative travel time between the refracted field and
the bottom interacting field is sufficiently small so as to not be
resolved by 1/3-octave processing at low frequencies. Thus, attempts
to measure the bottom-interacting field at these ranges may be

thwarted by the additional influence of the refracting field.

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To estimate this effect, a wave model which properly treats
caustic fields (the parabolic equation model mentioned earlier) was
run, including both the bottom-interacting and refracted fields,
and averaged in frequency over 1/3-octave at 35 Hz. The results of
these calculations are shown in Figure 15 with the rms sum, again, as
the estimated transmission loss. The inferred reflectivity is shown
at the top of the figure; it follows the earlier 35 Hz curve down to
about 5 degrees, below which it goes even further into negative values
than before. The significant point here is that even if the proper
summation of the bottom-interacting paths were used as the estimated
transmission loss, negative reflectivity losses would be obtained at
low grazing angles because of the refracted contribution to the field.
The effect of the refracted field will depend upon frequency, geometry

and depth excess.

To summarize the analysis at this point, it has been shown that
certain features of the low frequency bottom loss measurements made
by NOO and NADC, especially apparent negative bottom losses, could
be induced by 1) an over-simplified transmission loss model, and
2) inseparable bottom-interacting and refracted fields at ranges

corresponding to low grazing angles.

Bottom- reflected paths

So far the attention has focused on the model for propagation in
the water. Consider for a moment the diagram of Figure 16 which shows
not only a path refracting through the bottom, but one reflecting from
the boundary as well (some of the NOO data show the presence of both
paths). In general, if the incident amplitude is A and that of the
reflected path is aA, then the amplitude of the reflected path is
(1 - a)A (neglecting reflection back into the bottom of the emerging

refracted ray). When these paths recombine in the water, their

534

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

REFLECTIVITY LOSS (dB)

LOSS (dB)

100

Figure 15.

ANGLE (DEG)
Ss) 10 15 20

ACTUAL

INFERRED REFLECTIVITY

RMS SUM

SHOT PROCESSOR 35 Hz
(BOTTOM - BOUNCE &
CONVERGENCE ZONE)

BOTTOM - BOUNCE AND REFRACTED
TRANSMISSION LOSS

10 20 30 40
RANGE (NM)
REFLECTIVITY AND TRANSMISSION
LOSS FOR STRUCTURE OF FIGURE 14

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Figure 16. INTENSITY OF RECOMBINED SPLIT PATH

536

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

amplitude can be at most A (neglecting spreading losses) and, thus,
will not exceed the energy of a path for the case where only one of
these two is present. Such a mechanism, then, cannot give rise to

negative reflectivity.

While the above argument concludes that the presence of both
bottom-refracted and bottom-reflected paths cannot produce apparent
negative reflectivities, their simultaneous presence will certainly
produce interference patterns in the reflectivity as a function of
frequency. Figure 17 shows the relative arrival time structure for
the bottom-refracted and bottom-reflected paths for the case being
discussed here. Note that the reflected counterpart of each
refracted path arrives earlier; the fact that these paths arrive
simultaneously at maximum range is a direct consequence of no velocity
discontinuity at the bottom. The maximum travel-time difference
between these paths is about 15 msec for the geometry considered here;
the period of the corresponding variation with frequency of the
reflectivity will be 66 Hz or greater. Thus, the 1/3-octave filters
discussed above will give essentially the coherent combination of

these paths.

Finally, note that the need for dealing with four inseparable
paths could be avoided by getting the source and receiver away from
the ocean boundaries, but this in general will increase the inter-
ference of refracting fields. Conversely, the influence of the
refracting fields diminishes near the ocean surface, but the need
to deal with the four paths increases. This qualitative trade-off
suggests that low grazing angle measurements may always pose a

significant problem.

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Source Level Errors

Errors in the calibration of the source will directly affect the
inferred reflectivity. Worse, variations in the spectrum level of
the source from event to event will induce point-to-point errors in
the estimates of reflectivity. When pressing to get meaningful values
of reflectivity to accuracies of a few dB, it is obvious that all
measured or estimated qualities entering the computations must be
known to accuracies consistent with that desired in the result. Con-
cerns expressed elsewhere in these proceedings regarding measured
source levels for explosives have serious implications for measure-

ments, such as those here, which rely on absolute source levels.

CONCLUSION

This paper has attempted to develop a particular approach to the
design of transmission loss measurements based upon the use of exist-
ing acoustic models to estimate the nature of the loss in advance.
Examples were presented to illustrate common pitfalls which can be

avoided with this approach.

REFERENCES

Hanna, J. S., and B. E. Parkins, "Some considerations in choosing an
explosive source and processing filter for the measurement of
transmission loss," J. Acoust. Soc. Am. 56, 378-386, 1974

539

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

DISCUSSION

Dr. J. S. Hanna (Office of Naval Research): What I have here
(in Figure 16) is a schematic representation of a path which refracts
through the bottom and one which is reflected from the interface. What
I intended to have these paths correspond to is essentially what is back

at the source and again at the receiver.

The point that I wanted to make here is that even if both of
those paths are present in the data, the only thing that they can do
to you is produce structure in the bottom reflectivity that you would
infer, but they cannot give you reflectivities which are greater

than one.

The reason why, I believe, is that if these two paths have ampli-
tude A and they reach this boundary, the reflected path will have
some amplitude less than A which I have indicated by oA here. That
means that the energy, which is remaining to travel along this path,
is essentially 1 - a times the original amplitude, and the most that
can happen when these two recombine is that you get A back, but not

more than that and, in general, perhaps less than that.

So that I don't believe in principle that the combination of

those two paths is the problem.

Mr. M. A. Pedersen (Naval Undersea Center): No, that's not
quite true, because you have a slope discontinuity there, you are

bound to have a caustic down in that bottom medium. There is a slope

540

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

discontinuity at the interface because they have different velocity

gradients and it bends the right way to cause a caustic.

Dr. Hanna: The implication of that is what?

Mr. Pedersen: Well, that the intensity can be much larger for
this one that goes down into the bottom there than that other one.
Say, if the original energy that goes into the bottom is less than
the reflected ray, you could still get certain convergence regions

for that path that goes down next to bottom.

Dr. Hanna: Are you essentially saying that if I began with an
amplitude of A incident on the bottom and ran this path through the
problem and back out again — well, let me try to simplify the problem

just a little by ignoring the reflected path.

Let's say there is no reflection at the boundary and the only
thing that happens is that this path goes down, gets refracted and

comes out again.

Are you saying, essentially, that if I go to the surface here
where the path originates and terminates that I should expect to see
a received intensity for this path which may be higher than what

corresponds to simply keeping track of spreading loss along that path?

Mr. Pedersen: There is at least one more path. The point of
it is whenever you have one gradient, and then you have another
slope discontinuity to a steeper gradient, you always get a caustic,

if you increase the angle to a steep enough angle.

Mr. C. W. Spofford (Office of Naval Research): Yes, but, Mel,
that caustic is occurring way back in range in this problem. John is

talking of 5 or 10 degrees, and the caustic was around 25 or 30 degrees.

541

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Mr. J. I. Ewing (Lamont-Doherty Geological Observatory of
Columbia University): That's right, too. There is a critical
distance involved. In this situation there is a single ray path,
but beyond this distance there are two families. At an increased
range one family dives deeper into the bottom than the critical ray

does, and the other family goes shallower than the other ray does.

Beyond that critical distance you have two distinct paths.

One thing I object to is the neglect of sub-bottom reflections
in your treatment. Beyond the low grazing angle you are likely to
have rays reflecting off of sub-bottom interfaces at very favorable

angles of incidence to return the amount of energy.

Dr. Hanna: Okay, I should make it clear at this point that I
have, indeed, not included those possibilities in the problem and I
am not suggesting for a moment that they aren't out there in some

real case of interest to us.

Mr. Ewing: I agree when you are out there near the 30-mile
range you probably only have one refracting ray. The deeper ones
have probably already been intersected by either some sedimentary

reflector or by the basement rock.

Mr. Pedersen: It depends on where you cut off this positive
gradient layer there with your sedimentary bottom. But if you just
imagine continuing that on indefinitely, you see that branch has to

come back out again in range.

Dr. Hanna: That sounds like almost an academic thing to do,
though. That is to say, this sedimentary layer already als} X00) 5 15) 5\0)
fathoms deep and, in any event, I think it is unlikely that any
energy that penetrates this deeply, if there is any absorption in

the problem, is going to come back to haunt me again, anyway.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Mr. Ewing: Why was the reflected path from that sub-bottom inter-

face terminated where it was? I don't understand that.

Dr. Hanna: I'm sorry, which reflected path? You are talking

about the one which reflects at the interface itself?

Mr. Ewing: No, that reflects at the deeper interface.

Dr. Hanna: Oh, down here.

Mr. Ewing: Yes.

Dr. Hanna: In this particular case, they have simply come down

reflected and they do return into the problem, but way back in here.

Mr. Ewing: And then what?

Dr. Hanna: Well, for me, and then nothing, because I was inter-
ested in these ranges here and those correspond to the very short

ranges which I wasn't really discussing at this point.

Let me try to remember another one of Will's slides.

He showed measured transmission loss as a function of range for,

I guess it was — wasn't it — the Caribbean?
Mr. W. H. Geddes (Naval Oceanographic Office): That was the
Caribbean.

Dr. Hanna: Yes. And if you remember, going from long range
into decreasing range there was a very abrupt transition in trans-
mission loss at around — I don't know, it was around this range
right here, if I remember correctly, 24, 28 kiloyards, something
like that, where the transmission loss abruptly dropped and went on

back to the ranges corresponding to this part of the problem here

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

and there wasn't — at least the way I would read it — any indication
of contributions from these paths that might have reflected, let's

say, in that deep interface at all.

Dr. C. W. Horton, Sr. (Applied Research Laboratory of the
University of Texas at Austin): I think the point is that whether
there are two or more rays, your point is still proven. You are
wanting to show that you cannot get positive bottom-loss values,
and the total reflection, if there is no loss at the bottom, is
unity. So where there are two or three rays that add together,
you still would only get a total amplitude of unity if they are in

base. That is all you set out to establish.

Dr. Hanna: Yes, that's exactly right. It sounds like I have

convinced at least one person.

The point I wanted to make is, even if one improved this picture
to include for low grazing angles the possibility that some of the
incident energy is reflected and not refracted through this layer,
that you take that incident energy and send part of it along one path
and the remainder along the other path, and that some place in the
problem they may come together again. But the most that you can do
is get back to the original intensity of that path, less the spread-

ing loss.

Dr. J. B. Hersey (Office of Naval Research): John, have you
experimented with nonlinear gradients in the sediment? If there is
a second derivative to the gradient, I believe that caustic is
guaranteed, right? I can assure you this kind of intensification is
seen experimentally and it is very striking indeed. Its explanation

is illusive.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Suppose we have two rays that bend together repeatedly, you
know, successive rays forming a caustic, there can be intensifica-
tions without any violation or conservation of energy or anything

like that. It's commonly known as focusing.

All along, you see, we have had experimental evidence for years
that there is a strong intensification of an arrival that has to pass
through the bottom in some manner and arrive at ranges of the order

of 12-14 miles plus, and it continues strong for quite a few miles.

This is the experimental reason why I have believed in the
possibility of negative bottom loss. But I have been left very
hungry by these various ray analyses, because always it was carefully

explained to me that the velocity gradient in the bottom was linear.

Mr. Spofford: I think the point is we are after the plane wave
reflection coefficient of the bottom. This is what the models need,

this is why we are supposed to be out there measuring reflectivity.

If you go to a range where you think you are observing a 5 degree
grazing angle on the bottom and you are really seeing a reflected
angle at 5 degrees, plus an angle which is going into the bottom at
20 degrees, transiting through the sediment, and coming up again
with a strong focus (which is certainly possible if the sediment is
deep enough and the curvature is strong enough) you are not measuring
the reflectivity at 5 degrees, you are measuring transmission loss at

that point in range.

The point I think John and I are trying to get to is, there is
a specific mission in mind for these measurements which is bottom
reflectivity. If you put plane waves into the bottom at various

angles, you don't observe reflectivities greater than one.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Talking from a very parochial point of view, from the modeling
point of view, if you take a negative reflectivity and put it into a
propagation model and you go out a few hundred miles, adding in 2 or
3 dB increases of intensity per bounce for the grazing rays, you can
get your intensity to arbitrary levels at great range. They are

easy to do.

Dr. S. M. Flatte (University of California College of Santz Cruz):

Why can't you focus a plane wave?

Mr. Spofford: It will be focused on the bottom, but won't be

focused up above.

Dr. Flatte: No, it can be focused up above the bottom. It just
won't happen the next time. If you try and say it will do it many

times, it won't. But it can be focused the first time.

Mr. Spofford: The definition of the reflection coefficient
assumes that in a homogeneous medium we have an incident plane wave
of unit amplitude. Now, no matter what you put in the bottom, when
it comes out again, if you haven't put any absorption in the bottom,

it comes out with unit amplitude.

Dr. Hanna: JI have the same concern that Chuck has which is
that negative reflectivities are rather difficult for me to accommo-
date in any of the models I now have. That is not to say those models
should not learn how to accommodate to whatever those negative reflec-

tivities are trying to tell us.

The point that I want to make is that there are potential arti-
facts in some of those inferred bottom reflectivities produced by
the assumed transmission loss along the paths involved. I will feel

a lot more comfortable about debating the negative reflectivities and

546

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

after I think I have done as well as I possibly can toward modeling
the part of the propagation that is alledged to take place in the

water itself.

Mr. R. L. Martin (Naval Underwater Systems Center,New London
Laboratory): In the Labrador Basin, Stan Della has made some measure-
ments of an area of little interest to the Navy for bottom loss be-
cause there is so much depth excess, but he has been very careful to
separate out a direct path and a bottom reflected path. He has been
able to do this successfully down to angles of almost 5 degrees
grazing, and starting at 10 degrees he has observed what we call

"negative" bottom loss.

Even when you are very careful about your experimental procedure,
using shots that are detonated deep in the water column, the receiver
deep in the water column, and other factors, you still make that

observation of negative bottom loss.

The way models are used today, you can't throw that into a model
because every time the ray intersects the bottom, the negative loss is
put into that ray. But it does indicate that in those areas where we
make thet type of observation, that you perhaps have to include the
bottom into your model, because it is going to be a function of the

point in the water column where you make the measurement.

Dr. H. Weinberg (Naval Underwater Systems Center New London
Laboratory): Why don't we just simplify the problem and forget that
you even have a bottom. Just consider the ray that goes into a
little bottom region with a strong but positive gradient. By changing
that gradient, you can get just about any type of answer you want.
Clearly, by making that gradient strong enough, you can focus the

energy enough to get an increase in the power of its intensity.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

I think you are just arguing whether or not you interpret that

increase as a negative bottom loss or focusing.

It really doesn't matter what you call it, but it is possible

to get the same effect.

Dr. Hanna: The computations of intensity, that is, the rms sum
of those four paths that I showed, were made using a ray tracing
program which had that strong negative gradient in it. That is how
the rays were traced. It was not assumed that the rays reflected
from that bottom boundary, they refracted through it in that calcu-

lation.

Dr. Weinberg: Maybe in the particular example that you looked
at, what you are saying is exactly right. But it's easy enough to
construct another example where you can get focusing into that

bottom region.

Mr. Ewing: The way I look at it is that you have a gradient in
the water, and maybe you change the value of the gradient in the
sediment, but I think it can still even be linear, Brackett. The
simplest case is not to assume a discontinuity there. Consider an
infinitely thick section of sediment that just has a gradient, and

for the moment, let's forget about any possibility of reflection here.

I believe it is proper to say that you don't hear anything at
your receiver until you get to some critical distance from the
source, at which the value of the gradient permits a ray to be bent
around and get back to the surface. That happens at some specific

depth below the interface.

Then, if you imagine just one step beyond that, what you get

is one limiting ray that does not get quite as deep as the first one

548

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

did, and you get another ray that goes quite a lot deeper than that,

but coming to the same path.

I believe this is, in effect, your lens. When you also intro-
duce reflectors in the sub-bottom, you can, at the right angle of

incidence, produce, almost, unit reflectivity.

Dr. P. W. Smith (Bolt, Beranek, and Newman, Inc.): What we are
after is tabulated bottom losses for prediction of changes in the
environment which are independent of the bottom. We do not want
those changes in the environment independent of the bottom to change
the parameters by which we classify the bottom. This particular
focusing feature is peculiar to the environment. We want a charac-
terization of bottom reflection that will be useful for sampling.

The problem is a very complex one. How do we take this apart and get

a number that we can usefully use for transmission loss prediction.

Mr. Ewing: I fail to see how the energy that is returned from
the sediment is not part of the problem, because it is energy
returned into the water. A very large amount of it is returned to

the water.

Dr. I. Dyer (Department of Ocean Engineering of Massachusetts
Institute of Technology): The analogy might be that if for some
reason the model makers forced us to neglect the lower 2,000 meters
of the water column, and replaced the lower 2,000 meters of the water
column by an effective bottom water reflectivity and we find we have

convergent zones and we say, "Ooops! A negative reflection loss."

It seems to me that the problem here is no different. The
bottom is part of the column, and any attempt to put an artificial
line there and describe it by a simple number in this frequency

range is bound to fail.

549

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

So, the model makers are going to have to adjust. We can't

change the ocean.

Dr. Weinstein: John (Dr. Hanna), concerning another point, you
started out by saying, let's assume the source levels are properly
taken care of, but in fact they are not, because the measurement of
the source level is made at a relatively short range when you are

doing this kind of work.

At these ranges you are still in the shock wave region and not in
the pseudo-acoustic region. You find that the pressure-time curve
changes with range in such a fashion that there is a transfer of

energy from the high frequencies to the lows.

This, in itself, would give you an apparent negative bottom loss
if you apply spherical spreading as your means of correction, or if
you calculate the propagation loss assuming that you have a caustic

source.

Dr. Hanna: It is true that the analysis performed on this data
is more complicated than only worrying about the estimated loss that

you are going to compare to the measured transmission loss.

There is the whole problem of source level. I am not sure that
I would agree at this point that it is a mechanism for getting nega-
tive reflectivities except if the source level is too high or too

low, whichever way it has to be to make that happen.

I would like to make just one more comment about the particular
sound velocity structure that I used here and what rays are and are
not present at certain ranges in the problem for that particular

geometry.

550

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

One thing I can say with absolute certainty is that, setting
aside for the moment the question of whether you think the constant
gradient that I chose in the bottom is at all realistic, if you
accept the sound velocity structure, that I outlined above, to trace
rays, I can promise that in that particular case for the ranges from
14 to 30 miles, roughly, there is only one path which refracts in

that strong gradient then comes back at those ranges.

Now, the thing that I would certainly admit to the possibility
of is the following: It may be that for more complicated sound
velocity structures and for different values of, say, this initial
gradient, and the way that behaves with depth, that you can indeed
construct the kind of situations that you mentioned. That is, that
at the ranges I considered, you have steeper paths which come back

into the problem.

I would not quarrel with the possibility of doing that. The
only thing that I would maintain is that with this particular specific

example there is only that one path for those ranges.

What that may be telling us is that this example is not really

representative of most of the cases that you had in your experience.

Dr. M. Schulkin (Naval Oceanographic Office): You don't have a
negative bottom loss going continually out in range. It's just the
first one where there is an apparent gain over inverse square
spreading, because you have a convergence zone there. Like the other
convergent zones that you take for granted in the water column, you
have a 3 dB loss the distance level from there on, because you have

10 log R spreading as you continue down the path.

It is no violation of the conservation of energy. If you focus
your energy at some points, you lose it at other points in the vertical

column, say at that range.

Dil:

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

You don't deny this in regard to the series of convergent zones

systems?

Dr. Hanna: No. I think the only point I would make here is that
the caustic of this convergent zone occurs in the sediment, not back

up at the surface.

Dr. Schulkin: No, no. Just like your regular convergent zone.

You have got focusing near the surface of the regular convergent zone.

Dr. Hanna: Let me make just one further statement. Whatever
focusing is accomplished by this sound velocity structure should be
reflected in the ray tracing calculation; that is, the essence of
that calculation is to compute the spreading loss along that ray

along with whatever focusing the environment creates.

The curves that I showed you were made based on those kinds of
computations. So, in my construction, if there is any focusing along

that ray from whatever mechanism, it is in the computation.

It is in the curve that I call the rms sum of intensities which

was constructed from the computed intensity along each of those paths.

Dr. Schulkin: Phasing is very important and this rms combination
of your four rays — that's not what Will does anyway, as far as the
analysis goes, except for individual arrivais. I don't know why you
did that. The rms summation before arrival is not what Will analyzes

in his data.

Dr. Hanna: That is a very important point.

Dr. Hersey: I am going to take a chairman's privilege and
suggest that John go on to his next point and say that we have pin-

pointed a problem with which we had better deal.

552

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Dr. Hanna: JI have two things I want to say.

First, the main point that I was really trying to develop through-
out this whole discussion — not focusing on this specific example of
a kind of transmission loss measurement — is that when you make a
measurement of transmission loss and come back with a set of numbers,
in the processing of those data you need to think consciously about

what it is you believe you are measuring.

I can show examples where that kind of consideration has not
been given to the processing of the data, and the inferences drawn
from those data are, in fact, quite misleading. That is really the
essence of the point that I want to make. It certainly is not pro-
found to ask anybody who is doing something to think about what it

is they are doing.

Second, at this particular point, I am reminded of a story which

I think summarizes how I feel.

The way the story goes, a chicken and pig were riding in the
back of a farmer's truck. The truck was being driven through town.
The farmer hit a pothole in the road and the chicken and the pig
bounded out into the street. The truck went on, leaving them to

their own devices.

The chicken and the pig were strolling down the street at that
point and they passed a restaurant with the menu in the window: the

menu said, "Ham and eggs, $1.50."

The chicken looked at the sign, swelled up a bit with pride and
said to the pig, "Isn't it marvelous the contribution we make to
mankind." The pig looked at the chicken and said, "For you a

contribution, for me a personal sacrifice."

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Dr. Flatte: I want to tell Chuck that you can't focus a plane

wave, you are right.

Mr. Spofford: Thank you.

Mr. Pedersen: I have a few comments to make on the measurement
of bottom loss that might help resolve some of these ambiguous

situations.

When we make bottom-loss measurements, we perhaps run typically
out to the second or third convergent zone. We compare the loss per
bounce that we got by way of one bottom bounce, two bottom bounces,

and three bottom bounces.

That is, reduce all the data to a common base and then, if
these don't agree, you have a self-consistency check right on the
spot. That is, you don't have to come back another time to measure
to see if it was consistent to the extent that the bottom is uniform

over this distance.

You can make these comparisons and any errors in source level
always show up as a fixed displacement. That is, sometimes instead
of measuring the loss directly, you measure the difference between
the second bounce and the third bounce, or something like this. There

are certain fixed errors that can be removed in this fashion.

The second point about this is the problem of measuring loss at
the low angles as you approach the convergent zone. The relationship
of where the bottom reflected angle is intersected by the zone doesn't
stay constant from zone to zone. Generally speaking, if you have a
case where you just have surface reflected rays, I believe that you
can penetrate down to lower angles by going into the, say, third

convergent zone than by going into the second zone.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

Dr. Hanna: Following on that point, is it not true that Officer,
for example, applied exactly that kind of analysis to some of his low

frequency bottom reflectivity measurements?

Dr. Hersey: Several reflectivity measurements were made that

way.

I think they have a distinct fault, though. I have a funny
feeling in my stomach that they simply don't measure bottom loss,
and this has been growing on me through the evening. We did it

that way for several years.

Dr. Dyer: John, it may be a little late to come back to an

opening philosophical point that John made.

You said the motivation for measurements is to better understand
the ocean — I applaud that view. You said, also, the motivation for

measurement is to better build models — I applaud that view.

Who is going to speak for those poor guys who have to design
systems? It's a rather different kind of motivation. And how do

you design programs to meet those kinds of needs?

Dr. Hersey: Actually, Ira, I am somewhat disappointed that in
the main this first transient of our workshop hasn't addressed that

problem more than it has.

There is no question, however, that models based on a rational
consideration of the influence of the environment on acoustic propa-
gation and the shaping of the noise field have been applied to
estimating the performance of systems that have not been built on
analysis of performance of systems that have been built. The results
of the latter are very weak in resolving power because of the nature

of an operation exercise.

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HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

But those models have been able to account accurately — and by
accurately I mean the mean discrepancy of the order of a dB —
using the type of model we have been talking about so far in the

workshop.

However, I have to say that I have no evidence that we have done
justice to the systems designer. I guess he is going to have to tell
us precisely where he thinks we have fallen down, because we haven't

been able to account for the performance of this system.

Mr. Geddes: Regardless of how we process the data, we scale
these records. We find an arrival on the records the amplitude of
which is the largest thing on the record. It's there, record after

record after record.

So that, regardless of the explanation for it, I still have the
situation of looking at an arrival which I can look at on the records,

I can listen to it, and I can measure its amplitude.

Dr. Hersey: There is only one problem with what you just said,
Will. That is what is known as a disallowed area of concern. I

disallowed it about 15 minutes ago.

Dr. G. B. Morris (Marine Physical Laboratory, Scripps Institution
of Oceanography): I think use of models in planning experiments and

comparisons of the models with the experimental data have to be done.

Some of the examples that were shown are sort of extremes in that
you compare model data which have a very, very fine resolution with
experimental data which have a very long averaging time to very poor
resolution. It's the type of example that even experimentalists would
not think of doing, comparing a propagation curve that has, say,
values every few hundred yards, with another one that might have values

every few miles, except in a very gross manner.

556

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

I think one thing that we might want to do is to apply some
common filter functions to propagation loss models, so that the time
averaging used in the experimental data would correspond, say, fairly

closely to some sort of averaging in the model of propagation curves.

Dr. Hanna: I don't disagree with what Gerry said, I just make
the observation that for the two examples of CW data and CW calcula-
tions that I showed, the resolution in range, if you like, was com-

parable between the data and the calculation in both of those cases.

The first case that I showed represented a problem, if you like,
only because there were not enough experimental points with that
resolution. If those points had, say, been taken with an equivalent
range average of a mile or so, the model could have been run that
way and an interesting comparison made. But, unfortunately, given
the apparent underlying structure, you are faced with an under-sampled

curve and there wasn't a lot which you could do with it.

I don't mean to cast negatively on the experimentalists at that
particular point, but just to show that as an example of the kind
of difficulty that can arise without anticipating what the function

looks like that you are trying to measure.

Dr. Hersey: I should amend my comments, Ira, by saying, of course,
the models that we are talking about become considerably modified by
the addition to them of the punitive system characteristic. But we

have done that.
Dr. Dyer: We haven't talked much about that.

Dr. Hersey: You are dead right, and I am disappointed.

)s)7/

HANNA: DESIGN OF TRANSMISSION LOSS EXPERIMENTS

One observation, for example, we do most of our propagation
studies — not all, but most — with explosives. We have yet, I
think I am right, John — we have yet to model a single explosive
transmission event; isn't that correct? Don't we always make a CW
model and then sort of imagine that the CW model is like the

explosive?
Mr. Spofford: We are doing that.
Dr. Hersey: We have had this as a dream, I know.

Mr. Spofford: Of course with the ray models we can put in the
shot characteristics. So I would say at the moment we are a little
hard-pressed to come up with them exactly; this is the problem.

There is a linearity of something we do with shots.

558

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malt itl
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peace
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Full text of "International Workshop on Low-Frequency Propagation and Noise, Woods Hole, Massachusetts, 14-19 October, 1974"

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